Theorem 0xp 5438

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 4150	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
2		simprl 787	. . . . . 6 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ∅)
3	1, 2	mto 189	. . . . 5 ⊢ ¬ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
4	3	nex 1899	. . . 4 ⊢ ¬ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
5	4	nex 1899	. . 3 ⊢ ¬ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
6		elxp 5369	. . 3 ⊢ (𝑧 ∈ (∅ × 𝐴) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)))
7	5, 6	mtbir 315	. 2 ⊢ ¬ 𝑧 ∈ (∅ × 𝐴)
8	7	nel0 4163	1 ⊢ (∅ × 𝐴) = ∅

Syntax hints:   ∧ wa 386   = wceq 1656  ∃wex 1878   ∈ wcel 2164  ∅c0 4146  ⟨cop 4405   × cxp 5344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-opab 4938  df-xp 5352

This theorem is referenced by:  dmxpid  5581  csbres  5636  res0  5637  xp0  5797  xpnz  5798  xpdisj1  5800  difxp2  5805  xpcan2  5816  xpima  5821  unixp  5913  unixpid  5915  xpcoid  5921  fodomr  8386  xpfi  8506  cdaassen  9326  iundom2g  9684  alephadd  9721  hashxplem  13516  dmtrclfv  14143  ramcl  16111  0subcat  16857  mat0dimbas0  20647  mavmul0g  20734  txindislem  21814  txhaus  21828  tmdgsum  22276  ust0  22400  ehl0  23592  mbf0  23807  sibf0  30937  mexval2  31942  poimirlem5  33957  poimirlem10  33962  poimirlem22  33974  poimirlem23  33975  poimirlem26  33978  poimirlem28  33980  0heALT  38916