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Theorem 0xp 4666
Description: The cross 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.
StepHypRef Expression
1 elxp 4603 . . 3 (𝑧 ∈ (∅ × 𝐴) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴)))
2 noel 3398 . . . . . . 7 ¬ 𝑥 ∈ ∅
3 simprl 521 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴)) → 𝑥 ∈ ∅)
42, 3mto 652 . . . . . 6 ¬ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
54nex 1480 . . . . 5 ¬ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
65nex 1480 . . . 4 ¬ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
7 noel 3398 . . . 4 ¬ 𝑧 ∈ ∅
86, 72false 691 . . 3 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴)) ↔ 𝑧 ∈ ∅)
91, 8bitri 183 . 2 (𝑧 ∈ (∅ × 𝐴) ↔ 𝑧 ∈ ∅)
109eqriv 2154 1 (∅ × 𝐴) = ∅
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1335  wex 1472  wcel 2128  c0 3394  cop 3563   × cxp 4584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-opab 4026  df-xp 4592
This theorem is referenced by:  res0  4870  xp0  5005  xpeq0r  5008  xpdisj1  5010  xpima1  5032  xpfi  6874  exmidfodomrlemim  7136  hashxp  10700
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