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	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	phpeqd.2	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
	phpeqd.3	⊢ ( 𝜑 → 𝐴 ≈ 𝐵 )
Assertion	phpeqd	⊢ ( 𝜑 → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		phpeqd.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		phpeqd.2	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
3		phpeqd.3	⊢ ( 𝜑 → 𝐴 ≈ 𝐵 )
4	2	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → 𝐵 ⊆ 𝐴 )
5		simpr	⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → ¬ 𝐴 = 𝐵 )
6	5	neqcomd	⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → ¬ 𝐵 = 𝐴 )
7		dfpss2	⊢ ( 𝐵 ⊊ 𝐴 ↔ ( 𝐵 ⊆ 𝐴 ∧ ¬ 𝐵 = 𝐴 ) )
8	4 6 7	sylanbrc	⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → 𝐵 ⊊ 𝐴 )
9		php3	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ⊊ 𝐴 ) → 𝐵 ≺ 𝐴 )
10	1 8 9	syl2an2r	⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → 𝐵 ≺ 𝐴 )
11		sdomnen	⊢ ( 𝐵 ≺ 𝐴 → ¬ 𝐵 ≈ 𝐴 )
12		ensymfib	⊢ ( 𝐴 ∈ Fin → ( 𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴 ) )
13	12	notbid	⊢ ( 𝐴 ∈ Fin → ( ¬ 𝐴 ≈ 𝐵 ↔ ¬ 𝐵 ≈ 𝐴 ) )
14	13	biimpar	⊢ ( ( 𝐴 ∈ Fin ∧ ¬ 𝐵 ≈ 𝐴 ) → ¬ 𝐴 ≈ 𝐵 )
15	1 11 14	syl2an	⊢ ( ( 𝜑 ∧ 𝐵 ≺ 𝐴 ) → ¬ 𝐴 ≈ 𝐵 )
16	10 15	syldan	⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → ¬ 𝐴 ≈ 𝐵 )
17	16	ex	⊢ ( 𝜑 → ( ¬ 𝐴 = 𝐵 → ¬ 𝐴 ≈ 𝐵 ) )
18	3 17	mt4d	⊢ ( 𝜑 → 𝐴 = 𝐵 )