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Theorem 0xp 5775
Description: The Cartesian product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
0xp (∅ × 𝐴) = ∅

Proof of Theorem 0xp
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noel 4331 . . . . . 6 ¬ 𝑥 ∈ ∅
2 simprl 770 . . . . . 6 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴)) → 𝑥 ∈ ∅)
31, 2mto 196 . . . . 5 ¬ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
43nex 1803 . . . 4 ¬ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
54nex 1803 . . 3 ¬ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
6 elxp 5700 . . 3 (𝑧 ∈ (∅ × 𝐴) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴)))
75, 6mtbir 323 . 2 ¬ 𝑧 ∈ (∅ × 𝐴)
87nel0 4351 1 (∅ × 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wex 1782  wcel 2107  c0 4323  cop 4635   × cxp 5675
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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
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-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-xp 5683
This theorem is referenced by:  dmxpid  5930  csbres  5985  res0  5986  xp0  6158  xpnz  6159  xpdisj1  6161  difxp2  6166  xpcan2  6177  xpima  6182  unixp  6282  unixpid  6284  xpcoid  6290  fodomr  9128  xpfiOLD  9318  iundom2g  10535  hashxplem  14393  dmtrclfv  14965  ramcl  16962  0subcat  17788  mat0dimbas0  21968  mavmul0g  22055  txindislem  23137  txhaus  23151  tmdgsum  23599  ust0  23724  ehl0  24934  mbf0  25151  hashxpe  32019  gsumpart  32207  sibf0  33333  lpadlem3  33690  mexval2  34494  poimirlem5  36493  poimirlem10  36498  poimirlem22  36510  poimirlem23  36511  poimirlem26  36514  poimirlem28  36516  0no  42186  0heALT  42534
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