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Theorem ixp0x 8871
Description: An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)
Assertion
Ref Expression
ixp0x X𝑥 ∈ ∅ 𝐴 = {∅}

Proof of Theorem ixp0x
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 dfixp 8844 . 2 X𝑥 ∈ ∅ 𝐴 = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴)}
2 velsn 4607 . . . 4 (𝑓 ∈ {∅} ↔ 𝑓 = ∅)
3 fn0 6637 . . . 4 (𝑓 Fn ∅ ↔ 𝑓 = ∅)
4 ral0 4475 . . . . 5 𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴
54biantru 531 . . . 4 (𝑓 Fn ∅ ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴))
62, 3, 53bitr2i 299 . . 3 (𝑓 ∈ {∅} ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴))
76abbi2i 2874 . 2 {∅} = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴)}
81, 7eqtr4i 2768 1 X𝑥 ∈ ∅ 𝐴 = {∅}
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  {cab 2714  wral 3065  c0 4287  {csn 4591   Fn wfn 6496  cfv 6501  Xcixp 8842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-fun 6503  df-fn 6504  df-ixp 8843
This theorem is referenced by:  0elixp  8874  ptcmpfi  23180  finixpnum  36092  ioorrnopn  44620  ioorrnopnxr  44622  hoicvr  44863  ovnhoi  44918  ovnlecvr2  44925  hoiqssbl  44940  hoimbl  44946  iunhoiioo  44991
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