Theorem phphashrd

Description: Corollary of the Pigeonhole Principle using equality. Equivalent of phphashd with reversed arguments. (Contributed by Rohan Ridenour, 3-Aug-2023)

	Ref	Expression
Hypotheses	phphashrd.1	|- ( ph -> B e. Fin )
	phphashrd.2	|- ( ph -> A C_ B )
	phphashrd.3	|- ( ph -> ( # ` A ) = ( # ` B ) )
Assertion	phphashrd	|- ( ph -> A = B )

Proof

Step	Hyp	Ref	Expression
1		phphashrd.1	|- ( ph -> B e. Fin )
2		phphashrd.2	|- ( ph -> A C_ B )
3		phphashrd.3	|- ( ph -> ( # ` A ) = ( # ` B ) )
4	3	eqcomd	|- ( ph -> ( # ` B ) = ( # ` A ) )
5	1 2 4	phphashd	|- ( ph -> B = A )
6	5	eqcomd	|- ( ph -> A = B )