Theorem 0xp 5643

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 4295	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
2		simprl 769	. . . . . 6 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ∅)
3	1, 2	mto 199	. . . . 5 ⊢ ¬ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
4	3	nex 1797	. . . 4 ⊢ ¬ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
5	4	nex 1797	. . 3 ⊢ ¬ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
6		elxp 5572	. . 3 ⊢ (𝑧 ∈ (∅ × 𝐴) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)))
7	5, 6	mtbir 325	. 2 ⊢ ¬ 𝑧 ∈ (∅ × 𝐴)
8	7	nel0 4310	1 ⊢ (∅ × 𝐴) = ∅

Syntax hints:   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  ∅c0 4290  ⟨cop 4566   × cxp 5547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-opab 5121  df-xp 5555

This theorem is referenced by:  dmxpid  5794  csbres  5850  res0  5851  xp0  6009  xpnz  6010  xpdisj1  6012  difxp2  6017  xpcan2  6028  xpima  6033  unixp  6127  unixpid  6129  xpcoid  6135  fodomr  8662  xpfi  8783  iundom2g  9956  hashxplem  13788  dmtrclfv  14372  ramcl  16359  0subcat  17102  mat0dimbas0  21069  mavmul0g  21156  txindislem  22235  txhaus  22249  tmdgsum  22697  ust0  22822  ehl0  24014  mbf0  24229  hashxpe  30523  sibf0  31587  lpadlem3  31944  mexval2  32745  poimirlem5  34891  poimirlem10  34896  poimirlem22  34908  poimirlem23  34909  poimirlem26  34912  poimirlem28  34914  0heALT  40122