Theorem 0elixp 8948

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 5282	. . 3 ⊢ ∅ ∈ V
2	1	snid 4643	. 2 ⊢ ∅ ∈ {∅}
3		ixp0x 8945	. 2 ⊢ X𝑥 ∈ ∅ 𝐴 = {∅}
4	2, 3	eleqtrri 2834	1 ⊢ ∅ ∈ X𝑥 ∈ ∅ 𝐴

Syntax hints:   ∈ wcel 2109  ∅c0 4313  {csn 4606  Xcixp 8916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2540  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-fun 6538  df-fn 6539  df-ixp 8917

This theorem is referenced by: (None)