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Theorem 0xp 5189
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 3911 . . . . . 6 ¬ 𝑥 ∈ ∅
2 simprl 793 . . . . . 6 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴)) → 𝑥 ∈ ∅)
31, 2mto 188 . . . . 5 ¬ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
43nex 1729 . . . 4 ¬ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
54nex 1729 . . 3 ¬ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
6 elxp 5121 . . 3 (𝑧 ∈ (∅ × 𝐴) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴)))
75, 6mtbir 313 . 2 ¬ 𝑧 ∈ (∅ × 𝐴)
87nel0 3924 1 (∅ × 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1481  wex 1702  wcel 1988  c0 3907  cop 4174   × cxp 5102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-opab 4704  df-xp 5110
This theorem is referenced by:  dmxpid  5334  csbres  5388  res0  5389  xp0  5540  xpnz  5541  xpdisj1  5543  difxp2  5548  xpcan2  5559  xpima  5564  unixp  5656  unixpid  5658  xpcoid  5664  fodomr  8096  xpfi  8216  cdaassen  8989  iundom2g  9347  alephadd  9384  hashxplem  13203  dmtrclfv  13740  ramcl  15714  0subcat  16479  mat0dimbas0  20253  mavmul0g  20340  txindislem  21417  txhaus  21431  tmdgsum  21880  ust0  22004  sibf0  30370  mexval2  31374  poimirlem5  33385  poimirlem10  33390  poimirlem22  33402  poimirlem23  33403  poimirlem26  33406  poimirlem28  33408  0mbf  33426  0heALT  37897
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