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Theorem xpnz 6015
Description: The Cartesian product of nonempty classes is nonempty. (Variation of a theorem contributed by Raph Levien, 30-Jun-2006.) (Contributed by NM, 30-Jun-2006.)
Assertion
Ref Expression
xpnz ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅)

Proof of Theorem xpnz
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 4314 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 n0 4314 . . . . 5 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
31, 2anbi12i 626 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵))
4 exdistrv 1949 . . . 4 (∃𝑥𝑦(𝑥𝐴𝑦𝐵) ↔ (∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵))
53, 4bitr4i 279 . . 3 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ ∃𝑥𝑦(𝑥𝐴𝑦𝐵))
6 opex 5353 . . . . . 6 𝑥, 𝑦⟩ ∈ V
7 eleq1 2905 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (𝐴 × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
8 opelxp 5590 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
97, 8syl6bb 288 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵)))
106, 9spcev 3611 . . . . 5 ((𝑥𝐴𝑦𝐵) → ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
11 n0 4314 . . . . 5 ((𝐴 × 𝐵) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
1210, 11sylibr 235 . . . 4 ((𝑥𝐴𝑦𝐵) → (𝐴 × 𝐵) ≠ ∅)
1312exlimivv 1926 . . 3 (∃𝑥𝑦(𝑥𝐴𝑦𝐵) → (𝐴 × 𝐵) ≠ ∅)
145, 13sylbi 218 . 2 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (𝐴 × 𝐵) ≠ ∅)
15 xpeq1 5568 . . . . 5 (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵))
16 0xp 5648 . . . . 5 (∅ × 𝐵) = ∅
1715, 16syl6eq 2877 . . . 4 (𝐴 = ∅ → (𝐴 × 𝐵) = ∅)
1817necon3i 3053 . . 3 ((𝐴 × 𝐵) ≠ ∅ → 𝐴 ≠ ∅)
19 xpeq2 5575 . . . . 5 (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
20 xp0 6014 . . . . 5 (𝐴 × ∅) = ∅
2119, 20syl6eq 2877 . . . 4 (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
2221necon3i 3053 . . 3 ((𝐴 × 𝐵) ≠ ∅ → 𝐵 ≠ ∅)
2318, 22jca 512 . 2 ((𝐴 × 𝐵) ≠ ∅ → (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅))
2414, 23impbii 210 1 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2107  wne 3021  c0 4295  cop 4570   × cxp 5552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-xp 5560  df-rel 5561  df-cnv 5562
This theorem is referenced by:  xpeq0  6016  ssxpb  6030  xp11  6031  unixpid  6134  xpexr2  7617  frxp  7816  xpfir  8734  axcc2lem  9852  axdc4lem  9871  mamufacex  20935  txindis  22177  bj-xpnzex  34174  bj-1upln0  34224  bj-2upln1upl  34239  dibn0  38175
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