Theorem ixp0 8677

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 10170. (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 2946	. . . 4 ⊢ (¬ 𝐵 ≠ ∅ ↔ 𝐵 = ∅)
2	1	rexbii 3177	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝐵 ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝐵 = ∅)
3		rexnal 3165	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝐵 ≠ ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅)
4	2, 3	bitr3i 276	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅)
5		ixpn0 8676	. . 3 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅)
6	5	necon1bi 2971	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅)
7	4, 6	sylbi 216	1 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  ∅c0 4253  Xcixp 8643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-dif 3886  df-nul 4254  df-ixp 8644

This theorem is referenced by:  vonioo  44110  vonicc  44113