Theorem acnrcl

Description: Reverse closure for the choice set predicate. (Contributed by Mario Carneiro, 31-Aug-2015)

	Ref	Expression
Assertion	acnrcl	⊢ ( 𝑋 ∈ AC 𝐴 → 𝐴 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		ne0i	⊢ ( 𝑋 ∈ { 𝑥 ∣ ( 𝐴 ∈ V ∧ ∀ 𝑓 ∈ ( ( 𝒫 𝑥 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝑓 ‘ 𝑦 ) ) } → { 𝑥 ∣ ( 𝐴 ∈ V ∧ ∀ 𝑓 ∈ ( ( 𝒫 𝑥 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝑓 ‘ 𝑦 ) ) } ≠ ∅ )
2		abn0	⊢ ( { 𝑥 ∣ ( 𝐴 ∈ V ∧ ∀ 𝑓 ∈ ( ( 𝒫 𝑥 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝑓 ‘ 𝑦 ) ) } ≠ ∅ ↔ ∃ 𝑥 ( 𝐴 ∈ V ∧ ∀ 𝑓 ∈ ( ( 𝒫 𝑥 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝑓 ‘ 𝑦 ) ) )
3		simpl	⊢ ( ( 𝐴 ∈ V ∧ ∀ 𝑓 ∈ ( ( 𝒫 𝑥 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝑓 ‘ 𝑦 ) ) → 𝐴 ∈ V )
4	3	exlimiv	⊢ ( ∃ 𝑥 ( 𝐴 ∈ V ∧ ∀ 𝑓 ∈ ( ( 𝒫 𝑥 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝑓 ‘ 𝑦 ) ) → 𝐴 ∈ V )
5	2 4	sylbi	⊢ ( { 𝑥 ∣ ( 𝐴 ∈ V ∧ ∀ 𝑓 ∈ ( ( 𝒫 𝑥 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝑓 ‘ 𝑦 ) ) } ≠ ∅ → 𝐴 ∈ V )
6	1 5	syl	⊢ ( 𝑋 ∈ { 𝑥 ∣ ( 𝐴 ∈ V ∧ ∀ 𝑓 ∈ ( ( 𝒫 𝑥 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝑓 ‘ 𝑦 ) ) } → 𝐴 ∈ V )
7		df-acn	⊢ AC 𝐴 = { 𝑥 ∣ ( 𝐴 ∈ V ∧ ∀ 𝑓 ∈ ( ( 𝒫 𝑥 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝑓 ‘ 𝑦 ) ) }
8	6 7	eleq2s	⊢ ( 𝑋 ∈ AC 𝐴 → 𝐴 ∈ V )