Prisoner and Tiger behind the door puzzle

Prisoner and Tiger behind the door puzzle

A prisoner is facing three doors.
He is told that behind one of them, there is a lady,
and behind the other two there are tigers.
The sign on the door of the room containing the lady is true.
At least one of the other signs is false, maybe both.

The signs say
Room 1: A tiger is in room 2
Room 2: A tiger is in this room
Room 3 : A tiger is in room 1

Assuming that the prisoner will get eaten immediately
when opening the door to a room containing a tiger,
and moreover assuming that he wants to keep alive,
Which door should he open?

Answer :He should open Door to Room 1

Explanation :
Lady cannot be in room 2 since its sign says A tiger is in this room which will contradict the condition fo the sign being true of the room where the lady is.

So now the lady can be in room 1 or room 3
Assume the lady is in Room 1.
Then:
Room 1: A tiger is in room 2 - This is true
Room 2: A tiger is in this room - This becomes true
Room 3 : A tiger is in room 1 - This is False
So our assumption will meet all the conditions required if the lady is in room 1

Now lets assume the lady is in Room 3.
Then:
Room 1: A tiger is in room 2 - This is true
Room 2: A tiger is in this room - This is true
Room 3 : A tiger is in room 1 - This is true

This assumption of ours will not meet the conditions in the question.
Hence lady cannot be in room 3.

Therefore Lady has to be behind the door in room 1.
and hence the prisoner should open Door to Room 1.