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Theorem 0xp 4587
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 4524 . . 3 (𝑧 ∈ (∅ × 𝐴) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴)))
2 noel 3335 . . . . . . 7 ¬ 𝑥 ∈ ∅
3 simprl 503 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴)) → 𝑥 ∈ ∅)
42, 3mto 634 . . . . . 6 ¬ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
54nex 1459 . . . . 5 ¬ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
65nex 1459 . . . 4 ¬ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴))
7 noel 3335 . . . 4 ¬ 𝑧 ∈ ∅
86, 72false 673 . . 3 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦𝐴)) ↔ 𝑧 ∈ ∅)
91, 8bitri 183 . 2 (𝑧 ∈ (∅ × 𝐴) ↔ 𝑧 ∈ ∅)
109eqriv 2112 1 (∅ × 𝐴) = ∅
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1314  wex 1451  wcel 1463  c0 3331  cop 3498   × cxp 4505
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-opab 3958  df-xp 4513
This theorem is referenced by:  res0  4791  xp0  4926  xpeq0r  4929  xpdisj1  4931  xpima1  4953  xpfi  6784  exmidfodomrlemim  7021  hashxp  10523
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