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Theorem 0elixp 7825
 Description: Membership of the empty set in an infinite Cartesian product. (Contributed by Steve Rodriguez, 29-Sep-2006.)
Assertion
Ref Expression
0elixp ∅ ∈ X𝑥 ∈ ∅ 𝐴

Proof of Theorem 0elixp
StepHypRef Expression
1 0ex 4718 . . 3 ∅ ∈ V
21snid 4155 . 2 ∅ ∈ {∅}
3 ixp0x 7822 . 2 X𝑥 ∈ ∅ 𝐴 = {∅}
42, 3eleqtrri 2687 1 ∅ ∈ X𝑥 ∈ ∅ 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1977  ∅c0 3874  {csn 4125  Xcixp 7794 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-fun 5806  df-fn 5807  df-ixp 7795 This theorem is referenced by: (None)
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