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Theorem disjif 32592
Description: Property of a disjoint collection: if 𝐵(𝑥) and 𝐵(𝑌) = 𝐷 have a common element 𝑍, then 𝑥 = 𝑌. (Contributed by Thierry Arnoux, 30-Dec-2016.)
Hypotheses
Ref Expression
disjif.1 𝑥𝐶
disjif.2 (𝑥 = 𝑌𝐵 = 𝐶)
Assertion
Ref Expression
disjif ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝑍𝐵𝑍𝐶)) → 𝑥 = 𝑌)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑌
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑍(𝑥)

Proof of Theorem disjif
StepHypRef Expression
1 inelcm 4464 . 2 ((𝑍𝐵𝑍𝐶) → (𝐵𝐶) ≠ ∅)
2 disjif.1 . . . . . 6 𝑥𝐶
3 disjif.2 . . . . . 6 (𝑥 = 𝑌𝐵 = 𝐶)
42, 3disji2f 32591 . . . . 5 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ 𝑥𝑌) → (𝐵𝐶) = ∅)
543expia 1121 . . . 4 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → (𝑥𝑌 → (𝐵𝐶) = ∅))
65necon1d 2961 . . 3 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → ((𝐵𝐶) ≠ ∅ → 𝑥 = 𝑌))
763impia 1117 . 2 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝐵𝐶) ≠ ∅) → 𝑥 = 𝑌)
81, 7syl3an3 1165 1 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝑍𝐵𝑍𝐶)) → 𝑥 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  wnfc 2889  wne 2939  cin 3949  c0 4332  Disj wdisj 5109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rmo 3379  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-in 3957  df-nul 4333  df-disj 5110
This theorem is referenced by:  disjabrex  32596  2ndresdju  32660
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