Theorem ixp0 8970

Description: The infinite Cartesian product of a family 𝐵(𝑥) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 10521. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.)

Assertion

Ref	Expression
ixp0	⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅)

Proof of Theorem ixp0

Step	Hyp	Ref	Expression
1		nne 2942	. . . 4 ⊢ (¬ 𝐵 ≠ ∅ ↔ 𝐵 = ∅)
2	1	rexbii 3092	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝐵 ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝐵 = ∅)
3		rexnal 3098	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝐵 ≠ ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅)
4	2, 3	bitr3i 277	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅)
5		ixpn0 8969	. . 3 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅)
6	5	necon1bi 2967	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅)
7	4, 6	sylbi 217	1 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  ∅c0 4339  Xcixp 8936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-dif 3966  df-nul 4340  df-ixp 8937

This theorem is referenced by:  vonioo  46638  vonicc  46641