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Theorem 0xp 5233
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 3952 . . . . . 6 ¬ 𝑥 ∈ ∅
2 simprl 809 . . . . . 6 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴)) → 𝑥 ∈ ∅)
31, 2mto 188 . . . . 5 ¬ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
43nex 1771 . . . 4 ¬ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
54nex 1771 . . 3 ¬ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
6 elxp 5165 . . 3 (𝑧 ∈ (∅ × 𝐴) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴)))
75, 6mtbir 312 . 2 ¬ 𝑧 ∈ (∅ × 𝐴)
87nel0 3965 1 (∅ × 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wex 1744  wcel 2030  c0 3948  cop 4216   × cxp 5141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149
This theorem is referenced by:  dmxpid  5377  csbres  5431  res0  5432  xp0  5587  xpnz  5588  xpdisj1  5590  difxp2  5595  xpcan2  5606  xpima  5611  unixp  5706  unixpid  5708  xpcoid  5714  fodomr  8152  xpfi  8272  cdaassen  9042  iundom2g  9400  alephadd  9437  hashxplem  13258  dmtrclfv  13803  ramcl  15780  0subcat  16545  mat0dimbas0  20320  mavmul0g  20407  txindislem  21484  txhaus  21498  tmdgsum  21946  ust0  22070  sibf0  30524  mexval2  31526  poimirlem5  33544  poimirlem10  33549  poimirlem22  33561  poimirlem23  33562  poimirlem26  33565  poimirlem28  33567  0mbf  33585  0heALT  38394
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