Theorem 0xp 5720

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 4287	. . . . . 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 1801	. . . 4 ⊢ ¬ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
5	4	nex 1801	. . 3 ⊢ ¬ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
6		elxpi 5643	. . 3 ⊢ (𝑧 ∈ (∅ × 𝐴) → ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)))
7	5, 6	mto 197	. 2 ⊢ ¬ 𝑧 ∈ (∅ × 𝐴)
8	7	nel0 4303	1 ⊢ (∅ × 𝐴) = ∅

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∅c0 4282  ⟨cop 4583   × cxp 5619

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-dif 3901  df-nul 4283  df-opab 5158  df-xp 5627

This theorem is referenced by:  dmxpid  5876  csbres  5937  res0  5938  xp0OLD  6112  xpnz  6113  xpdisj1  6115  difxp2  6120  xpcan2  6131  xpima  6136  unixp  6236  unixpid  6238  xpcoid  6244  fodomr  9050  fodomfir  9221  iundom2g  10440  hashxplem  14344  dmtrclfv  14929  ramcl  16945  0subcat  17749  mat0dimbas0  22384  mavmul0g  22471  txindislem  23551  txhaus  23565  tmdgsum  24013  ust0  24138  ehl0  25347  mbf0  25565  fconst7v  32607  hashxpe  32796  indconst0  32848  indconst1  32849  gsumpart  33046  erlval  33234  fracbas  33280  sibf0  34370  lpadlem3  34714  mexval2  35570  poimirlem5  37688  poimirlem10  37693  poimirlem22  37705  poimirlem23  37706  poimirlem26  37709  poimirlem28  37711  0no  43555  0heALT  43903  dmrnxp  48964  0funcg2  49212  0funcALT  49216