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Theorem disjif 29277
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 4010 . 2 ((𝑍𝐵𝑍𝐶) → (𝐵𝐶) ≠ ∅)
2 disjif.1 . . . . . 6 𝑥𝐶
3 disjif.2 . . . . . 6 (𝑥 = 𝑌𝐵 = 𝐶)
42, 3disji2f 29276 . . . . 5 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ 𝑥𝑌) → (𝐵𝐶) = ∅)
543expia 1264 . . . 4 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → (𝑥𝑌 → (𝐵𝐶) = ∅))
65necon1d 2812 . . 3 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → ((𝐵𝐶) ≠ ∅ → 𝑥 = 𝑌))
763impia 1258 . 2 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝐵𝐶) ≠ ∅) → 𝑥 = 𝑌)
81, 7syl3an3 1358 1 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝑍𝐵𝑍𝐶)) → 𝑥 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wnfc 2748  wne 2790  cin 3559  c0 3897  Disj wdisj 4593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-in 3567  df-nul 3898  df-disj 4594
This theorem is referenced by:  disjabrex  29281
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