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Theorem disjex 29262
Description: Two ways to say that two classes are disjoint (or equal). (Contributed by Thierry Arnoux, 4-Oct-2016.)
Assertion
Ref Expression
disjex ((∃𝑧(𝑧𝐴𝑧𝐵) → 𝐴 = 𝐵) ↔ (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵

Proof of Theorem disjex
StepHypRef Expression
1 orcom 402 . 2 ((𝐴 = 𝐵 ∨ ¬ ∃𝑧(𝑧𝐴𝑧𝐵)) ↔ (¬ ∃𝑧(𝑧𝐴𝑧𝐵) ∨ 𝐴 = 𝐵))
2 df-in 3563 . . . . . 6 (𝐴𝐵) = {𝑧 ∣ (𝑧𝐴𝑧𝐵)}
32neeq1i 2854 . . . . 5 ((𝐴𝐵) ≠ ∅ ↔ {𝑧 ∣ (𝑧𝐴𝑧𝐵)} ≠ ∅)
4 abn0 3930 . . . . 5 ({𝑧 ∣ (𝑧𝐴𝑧𝐵)} ≠ ∅ ↔ ∃𝑧(𝑧𝐴𝑧𝐵))
53, 4bitr2i 265 . . . 4 (∃𝑧(𝑧𝐴𝑧𝐵) ↔ (𝐴𝐵) ≠ ∅)
65necon2bbii 2841 . . 3 ((𝐴𝐵) = ∅ ↔ ¬ ∃𝑧(𝑧𝐴𝑧𝐵))
76orbi2i 541 . 2 ((𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅) ↔ (𝐴 = 𝐵 ∨ ¬ ∃𝑧(𝑧𝐴𝑧𝐵)))
8 imor 428 . 2 ((∃𝑧(𝑧𝐴𝑧𝐵) → 𝐴 = 𝐵) ↔ (¬ ∃𝑧(𝑧𝐴𝑧𝐵) ∨ 𝐴 = 𝐵))
91, 7, 83bitr4ri 293 1 ((∃𝑧(𝑧𝐴𝑧𝐵) → 𝐴 = 𝐵) ↔ (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  cin 3555  c0 3893
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 3559  df-in 3563  df-nul 3894
This theorem is referenced by: (None)
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