Theorem 0elixp 8867

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 5252	. . 3 ⊢ ∅ ∈ V
2	1	snid 4619	. 2 ⊢ ∅ ∈ {∅}
3		ixp0x 8864	. 2 ⊢ X𝑥 ∈ ∅ 𝐴 = {∅}
4	2, 3	eleqtrri 2835	1 ⊢ ∅ ∈ X𝑥 ∈ ∅ 𝐴

Syntax hints:   ∈ wcel 2113  ∅c0 4285  {csn 4580  Xcixp 8835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-fun 6494  df-fn 6495  df-ixp 8836

This theorem is referenced by: (None)