Theorem 0xp 5710

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.

Step	Hyp	Ref	Expression
1		noel 4276	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
2		simprl 768	. . . . . 6 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ∅)
3	1, 2	mto 196	. . . . 5 ⊢ ¬ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
4	3	nex 1801	. . . 4 ⊢ ¬ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
5	4	nex 1801	. . 3 ⊢ ¬ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
6		elxp 5637	. . 3 ⊢ (𝑧 ∈ (∅ × 𝐴) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)))
7	5, 6	mtbir 322	. 2 ⊢ ¬ 𝑧 ∈ (∅ × 𝐴)
8	7	nel0 4296	1 ⊢ (∅ × 𝐴) = ∅

Syntax hints:   ∧ wa 396   = wceq 1540  ∃wex 1780   ∈ wcel 2105  ∅c0 4268  ⟨cop 4578   × cxp 5612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pr 5369

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-opab 5152  df-xp 5620

This theorem is referenced by:  dmxpid  5865  csbres  5920  res0  5921  xp0  6090  xpnz  6091  xpdisj1  6093  difxp2  6098  xpcan2  6109  xpima  6114  unixp  6214  unixpid  6216  xpcoid  6222  fodomr  8985  xpfiOLD  9175  iundom2g  10389  hashxplem  14240  dmtrclfv  14820  ramcl  16819  0subcat  17642  mat0dimbas0  21713  mavmul0g  21800  txindislem  22882  txhaus  22896  tmdgsum  23344  ust0  23469  ehl0  24679  mbf0  24896  hashxpe  31355  gsumpart  31543  sibf0  32542  lpadlem3  32899  mexval2  33705  poimirlem5  35880  poimirlem10  35885  poimirlem22  35897  poimirlem23  35898  poimirlem26  35901  poimirlem28  35903  0no  41352  0heALT  41701