Theorem 0xp 5758

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 4286	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
2		simprl 783	. . . . . 6 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ∅)
3	1, 2	mto 200	. . . . 5 ⊢ ¬ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
4	3	nex 1833	. . . 4 ⊢ ¬ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
5	4	nex 1833	. . 3 ⊢ ¬ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
6		elxpi 5680	. . 3 ⊢ (𝑧 ∈ (∅ × 𝐴) → ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)))
7	5, 6	mto 200	. 2 ⊢ ¬ 𝑧 ∈ (∅ × 𝐴)
8	7	nel0 4305	1 ⊢ (∅ × 𝐴) = ∅

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∅c0 4281  ⟨cop 4840   × cxp 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-dif 4050  df-nul 4282  df-opab 5174  df-xp 5664

This theorem is used by:  dmxpid  5924  csbres  5989  res0  5990  xp0OLD  6167  xpnz  6168  xpdisj1  6170  difxp2  6175  xpcan2  6187  xpima  6192  unixp  6309  unixpid  6311  xpcoid  6319  fodomr  9185  fodomfir  9357  iundom2g  10663  indconst0  12372  indconst1  12373  hashxplem  14619  dmtrclfv  15214  ramcl  17251  0subcat  18063  mat0dimbas0  22836  mavmul0g  22923  txindislem  24007  txhaus  24021  tmdgsum  24469  ust0  24594  ehl0  25793  mbf0  26010  fconst7v  33312  hashxpe  33495  gsumpart  33723  erlval  33918  fracbas  33971  0mplrim  34257  vieta  34323  sibf0  35077  lpadlem3  35421  mexval2  36476  poimirlem5  38811  poimirlem10  38816  poimirlem22  38828  poimirlem23  38829  poimirlem26  38832  poimirlem28  38834  0fno  44697  0heALT  45044  dmrnxp  50279  0funcg2  50530  0funcALT  50534