Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjexc Structured version   Visualization version   GIF version

Theorem disjexc 29251
Description: A variant of disjex 29250, 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 402 . . 3 ((𝐴 = 𝐵 ∨ ¬ ∃𝑧(𝑧𝐴𝑧𝐵)) ↔ (¬ ∃𝑧(𝑧𝐴𝑧𝐵) ∨ 𝐴 = 𝐵))
4 df-in 3562 . . . . . . 7 (𝐴𝐵) = {𝑧 ∣ (𝑧𝐴𝑧𝐵)}
54neeq1i 2854 . . . . . 6 ((𝐴𝐵) ≠ ∅ ↔ {𝑧 ∣ (𝑧𝐴𝑧𝐵)} ≠ ∅)
6 abn0 3928 . . . . . 6 ({𝑧 ∣ (𝑧𝐴𝑧𝐵)} ≠ ∅ ↔ ∃𝑧(𝑧𝐴𝑧𝐵))
75, 6bitr2i 265 . . . . 5 (∃𝑧(𝑧𝐴𝑧𝐵) ↔ (𝐴𝐵) ≠ ∅)
87necon2bbii 2841 . . . 4 ((𝐴𝐵) = ∅ ↔ ¬ ∃𝑧(𝑧𝐴𝑧𝐵))
98orbi2i 541 . . 3 ((𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅) ↔ (𝐴 = 𝐵 ∨ ¬ ∃𝑧(𝑧𝐴𝑧𝐵)))
10 imor 428 . . 3 ((∃𝑧(𝑧𝐴𝑧𝐵) → 𝐴 = 𝐵) ↔ (¬ ∃𝑧(𝑧𝐴𝑧𝐵) ∨ 𝐴 = 𝐵))
113, 9, 103bitr4i 292 . 2 ((𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅) ↔ (∃𝑧(𝑧𝐴𝑧𝐵) → 𝐴 = 𝐵))
122, 11sylibr 224 1 ((∃𝑧(𝑧𝐴𝑧𝐵) → 𝑥 = 𝑦) → (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  cin 3554  c0 3891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-v 3188  df-dif 3558  df-in 3562  df-nul 3892
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator