Theorem 0elixp 8905

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 5265	. . 3 ⊢ ∅ ∈ V
2	1	snid 4629	. 2 ⊢ ∅ ∈ {∅}
3		ixp0x 8902	. 2 ⊢ X𝑥 ∈ ∅ 𝐴 = {∅}
4	2, 3	eleqtrri 2828	1 ⊢ ∅ ∈ X𝑥 ∈ ∅ 𝐴

Syntax hints:   ∈ wcel 2109  ∅c0 4299  {csn 4592  Xcixp 8873

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-fun 6516  df-fn 6517  df-ixp 8874

This theorem is referenced by: (None)