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Theorem disjif 28773
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 3984 . 2 ((𝑍𝐵𝑍𝐶) → (𝐵𝐶) ≠ ∅)
2 disjif.1 . . . . . 6 𝑥𝐶
3 disjif.2 . . . . . 6 (𝑥 = 𝑌𝐵 = 𝐶)
42, 3disji2f 28772 . . . . 5 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ 𝑥𝑌) → (𝐵𝐶) = ∅)
543expia 1259 . . . 4 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → (𝑥𝑌 → (𝐵𝐶) = ∅))
65necon1d 2804 . . 3 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → ((𝐵𝐶) ≠ ∅ → 𝑥 = 𝑌))
763impia 1253 . 2 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝐵𝐶) ≠ ∅) → 𝑥 = 𝑌)
81, 7syl3an3 1353 1 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝑍𝐵𝑍𝐶)) → 𝑥 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wnfc 2738  wne 2780  cin 3539  c0 3874  Disj wdisj 4553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-in 3547  df-nul 3875  df-disj 4554
This theorem is referenced by:  disjabrex  28777
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