Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ndisj2 Structured version   Visualization version   GIF version

Theorem ndisj2 41190
Description: A non-disjointness condition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
ndisj2.1 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
ndisj2 Disj 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ (𝐵𝐶) ≠ ∅))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem ndisj2
StepHypRef Expression
1 ndisj2.1 . . . 4 (𝑥 = 𝑦𝐵 = 𝐶)
21disjor 5037 . . 3 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅))
32notbii 321 . 2 Disj 𝑥𝐴 𝐵 ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅))
4 rexnal 3235 . 2 (∃𝑥𝐴 ¬ ∀𝑦𝐴 (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅))
5 rexnal 3235 . . . 4 (∃𝑦𝐴 ¬ (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅) ↔ ¬ ∀𝑦𝐴 (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅))
6 ioran 977 . . . . . 6 (¬ (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅) ↔ (¬ 𝑥 = 𝑦 ∧ ¬ (𝐵𝐶) = ∅))
7 df-ne 3014 . . . . . . 7 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
8 df-ne 3014 . . . . . . 7 ((𝐵𝐶) ≠ ∅ ↔ ¬ (𝐵𝐶) = ∅)
97, 8anbi12i 626 . . . . . 6 ((𝑥𝑦 ∧ (𝐵𝐶) ≠ ∅) ↔ (¬ 𝑥 = 𝑦 ∧ ¬ (𝐵𝐶) = ∅))
106, 9bitr4i 279 . . . . 5 (¬ (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅) ↔ (𝑥𝑦 ∧ (𝐵𝐶) ≠ ∅))
1110rexbii 3244 . . . 4 (∃𝑦𝐴 ¬ (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅) ↔ ∃𝑦𝐴 (𝑥𝑦 ∧ (𝐵𝐶) ≠ ∅))
125, 11bitr3i 278 . . 3 (¬ ∀𝑦𝐴 (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅) ↔ ∃𝑦𝐴 (𝑥𝑦 ∧ (𝐵𝐶) ≠ ∅))
1312rexbii 3244 . 2 (∃𝑥𝐴 ¬ ∀𝑦𝐴 (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅) ↔ ∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ (𝐵𝐶) ≠ ∅))
143, 4, 133bitr2i 300 1 Disj 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ (𝐵𝐶) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wne 3013  wral 3135  wrex 3136  cin 3932  c0 4288  Disj wdisj 5022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rmo 3143  df-v 3494  df-dif 3936  df-in 3940  df-nul 4289  df-disj 5023
This theorem is referenced by:  disjrnmpt2  41325
  Copyright terms: Public domain W3C validator