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	$$⊢\phi \to A\in \mathrm{Fin}$$
	phpeqd.2	$$⊢\phi \to B\subseteq A$$
	phpeqd.3	$$⊢\phi \to A\approx B$$
Assertion	phpeqd	$$⊢\phi \to A=B$$

Proof

Step	Hyp	Ref	Expression
1		phpeqd.1	$$⊢\phi \to A\in \mathrm{Fin}$$
2		phpeqd.2	$$⊢\phi \to B\subseteq A$$
3		phpeqd.3	$$⊢\phi \to A\approx B$$
4	2	adantr	$$⊢\left(\phi \wedge \neg A=B\right)\to B\subseteq A$$
5		simpr	$$⊢\left(\phi \wedge \neg A=B\right)\to \neg A=B$$
6	5	neqcomd	$$⊢\left(\phi \wedge \neg A=B\right)\to \neg B=A$$
7		dfpss2	$$⊢B\subset A\leftrightarrow \left(B\subseteq A\wedge \neg B=A\right)$$
8	4 6 7	sylanbrc	$$⊢\left(\phi \wedge \neg A=B\right)\to B\subset A$$
9		php3	$$⊢\left(A\in \mathrm{Fin}\wedge B\subset A\right)\to B\prec A$$
10	1 8 9	syl2an2r	$$⊢\left(\phi \wedge \neg A=B\right)\to B\prec A$$
11		sdomnen	$$⊢B\prec A\to \neg B\approx A$$
12		ensymfib	$$⊢A\in \mathrm{Fin}\to \left(A\approx B\leftrightarrow B\approx A\right)$$
13	12	notbid	$$⊢A\in \mathrm{Fin}\to \left(\neg A\approx B\leftrightarrow \neg B\approx A\right)$$
14	13	biimpar	$$⊢\left(A\in \mathrm{Fin}\wedge \neg B\approx A\right)\to \neg A\approx B$$
15	1 11 14	syl2an	$$⊢\left(\phi \wedge B\prec A\right)\to \neg A\approx B$$
16	10 15	syldan	$$⊢\left(\phi \wedge \neg A=B\right)\to \neg A\approx B$$
17	16	ex	$$⊢\phi \to \left(\neg A=B\to \neg A\approx B\right)$$
18	3 17	mt4d	$$⊢\phi \to A=B$$