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

Theorem disjex 30399
 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 867 . 2 ((𝐴 = 𝐵 ∨ ¬ ∃𝑧(𝑧𝐴𝑧𝐵)) ↔ (¬ ∃𝑧(𝑧𝐴𝑧𝐵) ∨ 𝐴 = 𝐵))
2 df-in 3890 . . . . . 6 (𝐴𝐵) = {𝑧 ∣ (𝑧𝐴𝑧𝐵)}
32neeq1i 3051 . . . . 5 ((𝐴𝐵) ≠ ∅ ↔ {𝑧 ∣ (𝑧𝐴𝑧𝐵)} ≠ ∅)
4 abn0 4293 . . . . 5 ({𝑧 ∣ (𝑧𝐴𝑧𝐵)} ≠ ∅ ↔ ∃𝑧(𝑧𝐴𝑧𝐵))
53, 4bitr2i 279 . . . 4 (∃𝑧(𝑧𝐴𝑧𝐵) ↔ (𝐴𝐵) ≠ ∅)
65necon2bbii 3038 . . 3 ((𝐴𝐵) = ∅ ↔ ¬ ∃𝑧(𝑧𝐴𝑧𝐵))
76orbi2i 910 . 2 ((𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅) ↔ (𝐴 = 𝐵 ∨ ¬ ∃𝑧(𝑧𝐴𝑧𝐵)))
8 imor 850 . 2 ((∃𝑧(𝑧𝐴𝑧𝐵) → 𝐴 = 𝐵) ↔ (¬ ∃𝑧(𝑧𝐴𝑧𝐵) ∨ 𝐴 = 𝐵))
91, 7, 83bitr4ri 307 1 ((∃𝑧(𝑧𝐴𝑧𝐵) → 𝐴 = 𝐵) ↔ (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776   ≠ wne 2987   ∩ cin 3882  ∅c0 4246 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-dif 3886  df-in 3890  df-nul 4247 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator