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Theorem disjif 30914
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 4400 . 2 ((𝑍𝐵𝑍𝐶) → (𝐵𝐶) ≠ ∅)
2 disjif.1 . . . . . 6 𝑥𝐶
3 disjif.2 . . . . . 6 (𝑥 = 𝑌𝐵 = 𝐶)
42, 3disji2f 30913 . . . . 5 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ 𝑥𝑌) → (𝐵𝐶) = ∅)
543expia 1120 . . . 4 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → (𝑥𝑌 → (𝐵𝐶) = ∅))
65necon1d 2965 . . 3 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → ((𝐵𝐶) ≠ ∅ → 𝑥 = 𝑌))
763impia 1116 . 2 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝐵𝐶) ≠ ∅) → 𝑥 = 𝑌)
81, 7syl3an3 1164 1 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝑍𝐵𝑍𝐶)) → 𝑥 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  wnfc 2887  wne 2943  cin 3887  c0 4258  Disj wdisj 5041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rmo 3071  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-in 3895  df-nul 4259  df-disj 5042
This theorem is referenced by:  disjabrex  30918  2ndresdju  30983
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