Theorem 0xp 5772

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 4329	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
2		simprl 769	. . . . . 6 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ∅)
3	1, 2	mto 196	. . . . 5 ⊢ ¬ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
4	3	nex 1802	. . . 4 ⊢ ¬ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
5	4	nex 1802	. . 3 ⊢ ¬ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
6		elxp 5698	. . 3 ⊢ (𝑧 ∈ (∅ × 𝐴) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)))
7	5, 6	mtbir 322	. 2 ⊢ ¬ 𝑧 ∈ (∅ × 𝐴)
8	7	nel0 4349	1 ⊢ (∅ × 𝐴) = ∅

Syntax hints:   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∅c0 4321  ⟨cop 4633   × cxp 5673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-opab 5210  df-xp 5681

This theorem is referenced by:  dmxpid  5927  csbres  5982  res0  5983  xp0  6154  xpnz  6155  xpdisj1  6157  difxp2  6162  xpcan2  6173  xpima  6178  unixp  6278  unixpid  6280  xpcoid  6286  fodomr  9124  xpfiOLD  9314  iundom2g  10531  hashxplem  14389  dmtrclfv  14961  ramcl  16958  0subcat  17784  mat0dimbas0  21959  mavmul0g  22046  txindislem  23128  txhaus  23142  tmdgsum  23590  ust0  23715  ehl0  24925  mbf0  25142  hashxpe  32006  gsumpart  32194  sibf0  33321  lpadlem3  33678  mexval2  34482  poimirlem5  36481  poimirlem10  36486  poimirlem22  36498  poimirlem23  36499  poimirlem26  36502  poimirlem28  36504  0no  42171  0heALT  42519