Theorem phpeqd

Description: Corollary of the Pigeonhole Principle using equality. Strengthening of php expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023) Avoid ax-pow. (Revised by BTernaryTau, 28-Nov-2024)

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

Proof

Step	Hyp	Ref	Expression
1		phpeqd.1	|- ( ph -> A e. Fin )
2		phpeqd.2	|- ( ph -> B C_ A )
3		phpeqd.3	|- ( ph -> A ~~ B )
4	2	adantr	|- ( ( ph /\ -. A = B ) -> B C_ A )
5		simpr	|- ( ( ph /\ -. A = B ) -> -. A = B )
6	5	neqcomd	|- ( ( ph /\ -. A = B ) -> -. B = A )
7		dfpss2	|- ( B C. A <-> ( B C_ A /\ -. B = A ) )
8	4 6 7	sylanbrc	|- ( ( ph /\ -. A = B ) -> B C. A )
9		php3	|- ( ( A e. Fin /\ B C. A ) -> B ~< A )
10	1 8 9	syl2an2r	|- ( ( ph /\ -. A = B ) -> B ~< A )
11		sdomnen	|- ( B ~< A -> -. B ~~ A )
12		ensymfib	|- ( A e. Fin -> ( A ~~ B <-> B ~~ A ) )
13	12	notbid	|- ( A e. Fin -> ( -. A ~~ B <-> -. B ~~ A ) )
14	13	biimpar	|- ( ( A e. Fin /\ -. B ~~ A ) -> -. A ~~ B )
15	1 11 14	syl2an	|- ( ( ph /\ B ~< A ) -> -. A ~~ B )
16	10 15	syldan	|- ( ( ph /\ -. A = B ) -> -. A ~~ B )
17	16	ex	|- ( ph -> ( -. A = B -> -. A ~~ B ) )
18	3 17	mt4d	|- ( ph -> A = B )