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Theorem ndisj2 39717
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 4786 . . 3 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅))
32notbii 309 . 2 Disj 𝑥𝐴 𝐵 ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅))
4 rexnal 3133 . 2 (∃𝑥𝐴 ¬ ∀𝑦𝐴 (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅))
5 rexnal 3133 . . . 4 (∃𝑦𝐴 ¬ (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅) ↔ ¬ ∀𝑦𝐴 (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅))
6 ioran 512 . . . . . 6 (¬ (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅) ↔ (¬ 𝑥 = 𝑦 ∧ ¬ (𝐵𝐶) = ∅))
7 df-ne 2933 . . . . . . 7 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
8 df-ne 2933 . . . . . . 7 ((𝐵𝐶) ≠ ∅ ↔ ¬ (𝐵𝐶) = ∅)
97, 8anbi12i 735 . . . . . 6 ((𝑥𝑦 ∧ (𝐵𝐶) ≠ ∅) ↔ (¬ 𝑥 = 𝑦 ∧ ¬ (𝐵𝐶) = ∅))
106, 9bitr4i 267 . . . . 5 (¬ (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅) ↔ (𝑥𝑦 ∧ (𝐵𝐶) ≠ ∅))
1110rexbii 3179 . . . 4 (∃𝑦𝐴 ¬ (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅) ↔ ∃𝑦𝐴 (𝑥𝑦 ∧ (𝐵𝐶) ≠ ∅))
125, 11bitr3i 266 . . 3 (¬ ∀𝑦𝐴 (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅) ↔ ∃𝑦𝐴 (𝑥𝑦 ∧ (𝐵𝐶) ≠ ∅))
1312rexbii 3179 . 2 (∃𝑥𝐴 ¬ ∀𝑦𝐴 (𝑥 = 𝑦 ∨ (𝐵𝐶) = ∅) ↔ ∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ (𝐵𝐶) ≠ ∅))
143, 4, 133bitr2i 288 1 Disj 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ (𝐵𝐶) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wne 2932  wral 3050  wrex 3051  cin 3714  c0 4058  Disj wdisj 4772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rmo 3058  df-v 3342  df-dif 3718  df-in 3722  df-nul 4059  df-disj 4773
This theorem is referenced by:  disjrnmpt2  39874
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