Theorem 0elixp 8958

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 5312	. . 3 ⊢ ∅ ∈ V
2	1	snid 4669	. 2 ⊢ ∅ ∈ {∅}
3		ixp0x 8955	. 2 ⊢ X𝑥 ∈ ∅ 𝐴 = {∅}
4	2, 3	eleqtrri 2825	1 ⊢ ∅ ∈ X𝑥 ∈ ∅ 𝐴

Syntax hints:   ∈ wcel 2099  ∅c0 4325  {csn 4633  Xcixp 8926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-mo 2529  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-fun 6556  df-fn 6557  df-ixp 8927

This theorem is referenced by: (None)