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Metamath Proof Explorer

Theorem phphashd

Description: Corollary of the Pigeonhole Principle using equality. Equivalent of phpeqd expressed using the hash function. (Contributed by Rohan Ridenour, 3-Aug-2023)

Ref Expression
Hypotheses phphashd.1 ⊢ ( 𝜑 → 𝐴 ∈ Fin )
phphashd.2 ⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
phphashd.3 ⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) )
Assertion phphashd ( 𝜑 → 𝐴 = 𝐵 )

Proof

Step Hyp Ref Expression
1 phphashd.1 ⊢ ( 𝜑 → 𝐴 ∈ Fin )
2 phphashd.2 ⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
3 phphashd.3 ⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) )
4 1 2 ssfid ⊢ ( 𝜑 → 𝐵 ∈ Fin )
5 hashen ⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ) → ( ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) ↔ 𝐴 ≈ 𝐵 ) )
6 1 4 5 syl2anc ⊢ ( 𝜑 → ( ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) ↔ 𝐴 ≈ 𝐵 ) )
7 3 6 mpbid ⊢ ( 𝜑 → 𝐴 ≈ 𝐵 )
8 1 2 7 phpeqd ⊢ ( 𝜑 → 𝐴 = 𝐵 )