Theorem 0xp 5122

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		elxp 5055	. . 3 ⊢ (𝑧 ∈ (∅ × 𝐴) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)))
2		noel 3878	. . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅
3		simprl 790	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ∅)
4	2, 3	mto 187	. . . . . 6 ⊢ ¬ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
5	4	nex 1722	. . . . 5 ⊢ ¬ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
6	5	nex 1722	. . . 4 ⊢ ¬ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
7		noel 3878	. . . 4 ⊢ ¬ 𝑧 ∈ ∅
8	6, 7	2false 364	. . 3 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) ↔ 𝑧 ∈ ∅)
9	1, 8	bitri 263	. 2 ⊢ (𝑧 ∈ (∅ × 𝐴) ↔ 𝑧 ∈ ∅)
10	9	eqriv 2607	1 ⊢ (∅ × 𝐴) = ∅

Syntax hints:   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∅c0 3874  ⟨cop 4131   × cxp 5036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044

This theorem is referenced by:  dmxpid  5266  csbres  5320  res0  5321  xp0  5471  xpnz  5472  xpdisj1  5474  difxp2  5479  xpcan2  5490  xpima  5495  unixp  5585  unixpid  5587  xpcoid  5593  fodomr  7996  xpfi  8116  cdaassen  8887  iundom2g  9241  alephadd  9278  hashxplem  13080  dmtrclfv  13607  ramcl  15571  0subcat  16321  mat0dimbas0  20091  mavmul0g  20178  txindislem  21246  txhaus  21260  tmdgsum  21709  ust0  21833  sibf0  29723  mexval2  30654  poimirlem5  32584  poimirlem10  32589  poimirlem22  32601  poimirlem23  32602  poimirlem26  32605  poimirlem28  32607  0mbf  32625  0heALT  37097