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Metamath Proof Explorer

Theorem tz6.12-2OLD

Description: Obsolete version of tz6.12-2 as of 25-Jan-2026. (Contributed by NM, 30-Apr-2004) (Proof shortened by Mario Carneiro, 31-Aug-2015) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	tz6.12-2OLD	⊢ ( ¬ ∃! 𝑥 𝐴 𝐹 𝑥 → ( 𝐹 ‘ 𝐴 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		df-fv	⊢ ( 𝐹 ‘ 𝐴 ) = ( ℩ 𝑥 𝐴 𝐹 𝑥 )
2		iotanul	⊢ ( ¬ ∃! 𝑥 𝐴 𝐹 𝑥 → ( ℩ 𝑥 𝐴 𝐹 𝑥 ) = ∅ )
3	1 2	eqtrid	⊢ ( ¬ ∃! 𝑥 𝐴 𝐹 𝑥 → ( 𝐹 ‘ 𝐴 ) = ∅ )