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Metamath Proof Explorer

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 )