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Theorem disjexc 30259
Description: A variant of disjex 30258, applicable for more generic families. (Contributed by Thierry Arnoux, 4-Oct-2016.)
Hypothesis
Ref Expression
disjexc.1 (𝑥 = 𝑦𝐴 = 𝐵)
Assertion
Ref Expression
disjexc ((∃𝑧(𝑧𝐴𝑧𝐵) → 𝑥 = 𝑦) → (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem disjexc
StepHypRef Expression
1 disjexc.1 . . 3 (𝑥 = 𝑦𝐴 = 𝐵)
21imim2i 16 . 2 ((∃𝑧(𝑧𝐴𝑧𝐵) → 𝑥 = 𝑦) → (∃𝑧(𝑧𝐴𝑧𝐵) → 𝐴 = 𝐵))
3 orcom 866 . . 3 ((𝐴 = 𝐵 ∨ ¬ ∃𝑧(𝑧𝐴𝑧𝐵)) ↔ (¬ ∃𝑧(𝑧𝐴𝑧𝐵) ∨ 𝐴 = 𝐵))
4 df-in 3946 . . . . . . 7 (𝐴𝐵) = {𝑧 ∣ (𝑧𝐴𝑧𝐵)}
54neeq1i 3084 . . . . . 6 ((𝐴𝐵) ≠ ∅ ↔ {𝑧 ∣ (𝑧𝐴𝑧𝐵)} ≠ ∅)
6 abn0 4339 . . . . . 6 ({𝑧 ∣ (𝑧𝐴𝑧𝐵)} ≠ ∅ ↔ ∃𝑧(𝑧𝐴𝑧𝐵))
75, 6bitr2i 277 . . . . 5 (∃𝑧(𝑧𝐴𝑧𝐵) ↔ (𝐴𝐵) ≠ ∅)
87necon2bbii 3071 . . . 4 ((𝐴𝐵) = ∅ ↔ ¬ ∃𝑧(𝑧𝐴𝑧𝐵))
98orbi2i 908 . . 3 ((𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅) ↔ (𝐴 = 𝐵 ∨ ¬ ∃𝑧(𝑧𝐴𝑧𝐵)))
10 imor 849 . . 3 ((∃𝑧(𝑧𝐴𝑧𝐵) → 𝐴 = 𝐵) ↔ (¬ ∃𝑧(𝑧𝐴𝑧𝐵) ∨ 𝐴 = 𝐵))
113, 9, 103bitr4i 304 . 2 ((𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅) ↔ (∃𝑧(𝑧𝐴𝑧𝐵) → 𝐴 = 𝐵))
122, 11sylibr 235 1 ((∃𝑧(𝑧𝐴𝑧𝐵) → 𝑥 = 𝑦) → (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 843   = wceq 1530  wex 1773  wcel 2106  {cab 2802  wne 3020  cin 3938  c0 4294
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 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-dif 3942  df-in 3946  df-nul 4295
This theorem is referenced by: (None)
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