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Theorem disjexc 32613
Description: A variant of disjex 32612, 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 870 . . 3 ((𝐴 = 𝐵 ∨ ¬ ∃𝑧(𝑧𝐴𝑧𝐵)) ↔ (¬ ∃𝑧(𝑧𝐴𝑧𝐵) ∨ 𝐴 = 𝐵))
4 df-in 3970 . . . . . . 7 (𝐴𝐵) = {𝑧 ∣ (𝑧𝐴𝑧𝐵)}
54neeq1i 3003 . . . . . 6 ((𝐴𝐵) ≠ ∅ ↔ {𝑧 ∣ (𝑧𝐴𝑧𝐵)} ≠ ∅)
6 abn0 4391 . . . . . 6 ({𝑧 ∣ (𝑧𝐴𝑧𝐵)} ≠ ∅ ↔ ∃𝑧(𝑧𝐴𝑧𝐵))
75, 6bitr2i 276 . . . . 5 (∃𝑧(𝑧𝐴𝑧𝐵) ↔ (𝐴𝐵) ≠ ∅)
87necon2bbii 2990 . . . 4 ((𝐴𝐵) = ∅ ↔ ¬ ∃𝑧(𝑧𝐴𝑧𝐵))
98orbi2i 912 . . 3 ((𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅) ↔ (𝐴 = 𝐵 ∨ ¬ ∃𝑧(𝑧𝐴𝑧𝐵)))
10 imor 853 . . 3 ((∃𝑧(𝑧𝐴𝑧𝐵) → 𝐴 = 𝐵) ↔ (¬ ∃𝑧(𝑧𝐴𝑧𝐵) ∨ 𝐴 = 𝐵))
113, 9, 103bitr4i 303 . 2 ((𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅) ↔ (∃𝑧(𝑧𝐴𝑧𝐵) → 𝐴 = 𝐵))
122, 11sylibr 234 1 ((∃𝑧(𝑧𝐴𝑧𝐵) → 𝑥 = 𝑦) → (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wne 2938  cin 3962  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-ne 2939  df-dif 3966  df-in 3970  df-nul 4340
This theorem is referenced by: (None)
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