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Theorem disjexc 28788
 Description: A variant of disjex 28787, 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 401 . . 3 ((𝐴 = 𝐵 ∨ ¬ ∃𝑧(𝑧𝐴𝑧𝐵)) ↔ (¬ ∃𝑧(𝑧𝐴𝑧𝐵) ∨ 𝐴 = 𝐵))
4 df-in 3547 . . . . . . 7 (𝐴𝐵) = {𝑧 ∣ (𝑧𝐴𝑧𝐵)}
54neeq1i 2846 . . . . . 6 ((𝐴𝐵) ≠ ∅ ↔ {𝑧 ∣ (𝑧𝐴𝑧𝐵)} ≠ ∅)
6 abn0 3908 . . . . . 6 ({𝑧 ∣ (𝑧𝐴𝑧𝐵)} ≠ ∅ ↔ ∃𝑧(𝑧𝐴𝑧𝐵))
75, 6bitr2i 264 . . . . 5 (∃𝑧(𝑧𝐴𝑧𝐵) ↔ (𝐴𝐵) ≠ ∅)
87necon2bbii 2833 . . . 4 ((𝐴𝐵) = ∅ ↔ ¬ ∃𝑧(𝑧𝐴𝑧𝐵))
98orbi2i 540 . . 3 ((𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅) ↔ (𝐴 = 𝐵 ∨ ¬ ∃𝑧(𝑧𝐴𝑧𝐵)))
10 imor 427 . . 3 ((∃𝑧(𝑧𝐴𝑧𝐵) → 𝐴 = 𝐵) ↔ (¬ ∃𝑧(𝑧𝐴𝑧𝐵) ∨ 𝐴 = 𝐵))
113, 9, 103bitr4i 291 . 2 ((𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅) ↔ (∃𝑧(𝑧𝐴𝑧𝐵) → 𝐴 = 𝐵))
122, 11sylibr 223 1 ((∃𝑧(𝑧𝐴𝑧𝐵) → 𝑥 = 𝑦) → (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780   ∩ cin 3539  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-in 3547  df-nul 3875 This theorem is referenced by: (None)
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