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Theorem ixp0 7893
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 9257. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.)
Assertion
Ref Expression
ixp0 (∃𝑥𝐴 𝐵 = ∅ → X𝑥𝐴 𝐵 = ∅)

Proof of Theorem ixp0
StepHypRef Expression
1 nne 2794 . . . 4 𝐵 ≠ ∅ ↔ 𝐵 = ∅)
21rexbii 3035 . . 3 (∃𝑥𝐴 ¬ 𝐵 ≠ ∅ ↔ ∃𝑥𝐴 𝐵 = ∅)
3 rexnal 2990 . . 3 (∃𝑥𝐴 ¬ 𝐵 ≠ ∅ ↔ ¬ ∀𝑥𝐴 𝐵 ≠ ∅)
42, 3bitr3i 266 . 2 (∃𝑥𝐴 𝐵 = ∅ ↔ ¬ ∀𝑥𝐴 𝐵 ≠ ∅)
5 ixpn0 7892 . . 3 (X𝑥𝐴 𝐵 ≠ ∅ → ∀𝑥𝐴 𝐵 ≠ ∅)
65necon1bi 2818 . 2 (¬ ∀𝑥𝐴 𝐵 ≠ ∅ → X𝑥𝐴 𝐵 = ∅)
74, 6sylbi 207 1 (∃𝑥𝐴 𝐵 = ∅ → X𝑥𝐴 𝐵 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  wne 2790  wral 2907  wrex 2908  c0 3896  Xcixp 7860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-v 3191  df-dif 3562  df-nul 3897  df-ixp 7861
This theorem is referenced by:  vonioo  40229  vonicc  40232
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