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Metamath Proof Explorer

Theorem acacni

Description: A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015)

Ref Expression
Assertion acacni ( ( CHOICE ∧ 𝐴 ∈ 𝑉 ) → AC 𝐴 = V )

Proof

Step Hyp Ref Expression
1 simpr ⊢ ( ( CHOICE ∧ 𝐴 ∈ 𝑉 ) → 𝐴 ∈ 𝑉 )
2 vex ⊢ 𝑥 ∈ V
3 simpl ⊢ ( ( CHOICE ∧ 𝐴 ∈ 𝑉 ) → CHOICE )
4 dfac10 ⊢ ( CHOICE ↔ dom card = V )
5 3 4 sylib ⊢ ( ( CHOICE ∧ 𝐴 ∈ 𝑉 ) → dom card = V )
6 2 5 eleqtrrid ⊢ ( ( CHOICE ∧ 𝐴 ∈ 𝑉 ) → 𝑥 ∈ dom card )
7 numacn ⊢ ( 𝐴 ∈ 𝑉 → ( 𝑥 ∈ dom card → 𝑥 ∈ AC 𝐴 ) )
8 1 6 7 sylc ⊢ ( ( CHOICE ∧ 𝐴 ∈ 𝑉 ) → 𝑥 ∈ AC 𝐴 )
9 2 a1i ⊢ ( ( CHOICE ∧ 𝐴 ∈ 𝑉 ) → 𝑥 ∈ V )
10 8 9 2thd ⊢ ( ( CHOICE ∧ 𝐴 ∈ 𝑉 ) → ( 𝑥 ∈ AC 𝐴 ↔ 𝑥 ∈ V ) )
11 10 eqrdv ⊢ ( ( CHOICE ∧ 𝐴 ∈ 𝑉 ) → AC 𝐴 = V )