Theorem ixp0 8925

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 10462. (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 2962	. . . 4 ⊢ (¬ 𝐵 ≠ ∅ ↔ 𝐵 = ∅)
2	1	rexbii 3112	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝐵 ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝐵 = ∅)
3		rexnal 3117	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝐵 ≠ ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅)
4	2, 3	bitr3i 280	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅)
5		ixpn0 8924	. . 3 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅)
6	5	necon1bi 2986	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅)
7	4, 6	sylbi 220	1 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1570   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  ∅c0 4286  Xcixp 8891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-dif 3908  df-nul 4287  df-ixp 8892

This theorem is referenced by:  vonioo  47396  vonicc  47399