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Theorem 0xp 4488
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 4430 . . 3 (𝑧 ∈ (∅ × 𝐴) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴)))
2 noel 3279 . . . . . . 7 ¬ 𝑥 ∈ ∅
3 simprl 498 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴)) → 𝑥 ∈ ∅)
42, 3mto 621 . . . . . 6 ¬ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
54nex 1432 . . . . 5 ¬ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
65nex 1432 . . . 4 ¬ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
7 noel 3279 . . . 4 ¬ 𝑧 ∈ ∅
86, 72false 650 . . 3 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴)) ↔ 𝑧 ∈ ∅)
91, 8bitri 182 . 2 (𝑧 ∈ (∅ × 𝐴) ↔ 𝑧 ∈ ∅)
109eqriv 2082 1 (∅ × 𝐴) = ∅
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1287  wex 1424  wcel 1436  c0 3275  cop 3434   × cxp 4411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-opab 3877  df-xp 4419
This theorem is referenced by:  res0  4687  xp0  4819  xpeq0r  4822  xpdisj1  4823  xpima1  4845  xpfi  6593  exmidfodomrlemim  6774  hashxp  10134
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