Theorem 0elixp 8923

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

Step	Hyp	Ref	Expression
1		0ex 5269	. . 3 ⊢ ∅ ∈ V
2	1	snid 4630	. 2 ⊢ ∅ ∈ {∅}
3		ixp0x 8920	. 2 ⊢ X𝑥 ∈ ∅ 𝐴 = {∅}
4	2, 3	eleqtrri 2868	1 ⊢ ∅ ∈ X𝑥 ∈ ∅ 𝐴

Syntax hints:   ∈ wcel 2149  ∅c0 4294  {csn 4591  Xcixp 8891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-fun 6535  df-fn 6536  df-ixp 8892

This theorem is referenced by: (None)