Theorem ixp0 8879

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 10405. (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 2936	. . . 4 ⊢ (¬ 𝐵 ≠ ∅ ↔ 𝐵 = ∅)
2	1	rexbii 3084	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝐵 ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝐵 = ∅)
3		rexnal 3089	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝐵 ≠ ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅)
4	2, 3	bitr3i 277	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅)
5		ixpn0 8878	. . 3 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅)
6	5	necon1bi 2960	. 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 1542   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  ∅c0 4273  Xcixp 8845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-dif 3892  df-nul 4274  df-ixp 8846

This theorem is referenced by:  vonioo  47110  vonicc  47113