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Theorem 0xp 5642
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 4294 . . . . . 6 ¬ 𝑥 ∈ ∅
2 simprl 769 . . . . . 6 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴)) → 𝑥 ∈ ∅)
31, 2mto 199 . . . . 5 ¬ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
43nex 1794 . . . 4 ¬ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
54nex 1794 . . 3 ¬ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
6 elxp 5571 . . 3 (𝑧 ∈ (∅ × 𝐴) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴)))
75, 6mtbir 325 . 2 ¬ 𝑧 ∈ (∅ × 𝐴)
87nel0 4309 1 (∅ × 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 398   = wceq 1530  wex 1773  wcel 2107  c0 4289  cop 4565   × cxp 5546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-opab 5120  df-xp 5554
This theorem is referenced by:  dmxpid  5793  csbres  5849  res0  5850  xp0  6008  xpnz  6009  xpdisj1  6011  difxp2  6016  xpcan2  6027  xpima  6032  unixp  6126  unixpid  6128  xpcoid  6134  fodomr  8660  xpfi  8781  iundom2g  9954  hashxplem  13786  dmtrclfv  14370  ramcl  16357  0subcat  17100  mat0dimbas0  21067  mavmul0g  21154  txindislem  22233  txhaus  22247  tmdgsum  22695  ust0  22820  ehl0  24012  mbf0  24227  hashxpe  30521  sibf0  31585  lpadlem3  31942  mexval2  32743  poimirlem5  34889  poimirlem10  34894  poimirlem22  34906  poimirlem23  34907  poimirlem26  34910  poimirlem28  34912  0heALT  40119
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