Theorem 0xp 5753

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 4313	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
2		simprl 770	. . . . . 6 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ∅)
3	1, 2	mto 197	. . . . 5 ⊢ ¬ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
4	3	nex 1800	. . . 4 ⊢ ¬ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
5	4	nex 1800	. . 3 ⊢ ¬ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
6		elxp 5677	. . 3 ⊢ (𝑧 ∈ (∅ × 𝐴) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)))
7	5, 6	mtbir 323	. 2 ⊢ ¬ 𝑧 ∈ (∅ × 𝐴)
8	7	nel0 4329	1 ⊢ (∅ × 𝐴) = ∅

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∅c0 4308  ⟨cop 4607   × cxp 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-opab 5182  df-xp 5660

This theorem is referenced by:  dmxpid  5910  csbres  5969  res0  5970  xp0  6147  xpnz  6148  xpdisj1  6150  difxp2  6155  xpcan2  6166  xpima  6171  unixp  6271  unixpid  6273  xpcoid  6279  fodomr  9140  xpfiOLD  9329  fodomfir  9338  iundom2g  10552  hashxplem  14449  dmtrclfv  15035  ramcl  17047  0subcat  17849  mat0dimbas0  22402  mavmul0g  22489  txindislem  23569  txhaus  23583  tmdgsum  24031  ust0  24156  ehl0  25367  mbf0  25585  hashxpe  32732  gsumpart  32997  erlval  33199  fracbas  33245  sibf0  34312  lpadlem3  34656  mexval2  35471  poimirlem5  37595  poimirlem10  37600  poimirlem22  37612  poimirlem23  37613  poimirlem26  37616  poimirlem28  37618  0no  43406  0heALT  43754  dmrnxp  48763  0funcg2  48997  0funcALT  49001