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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	$⊢ φ \to A \in Fin$
	phphashd.2	$⊢ φ \to B \subseteq A$
	phphashd.3	$⊢ φ \to \|A\| = \|B\|$
Assertion	phphashd	$⊢ φ \to A = B$

Proof

Step	Hyp	Ref	Expression
1		phphashd.1	$⊢ φ \to A \in Fin$
2		phphashd.2	$⊢ φ \to B \subseteq A$
3		phphashd.3	$⊢ φ \to \|A\| = \|B\|$
4	1 2	ssfid	$⊢ φ \to B \in Fin$
5		hashen	$⊢ (A \in Fin \land B \in Fin) \to (\|A\| = \|B\| \leftrightarrow A \approx B)$
6	1 4 5	syl2anc	$⊢ φ \to (\|A\| = \|B\| \leftrightarrow A \approx B)$
7	3 6	mpbid	$⊢ φ \to A \approx B$
8	1 2 7	phpeqd	$⊢ φ \to A = B$