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Theorem disji 4570
 Description: Property of a disjoint collection: if 𝐵(𝑋) = 𝐶 and 𝐵(𝑌) = 𝐷 have a common element 𝑍, then 𝑋 = 𝑌. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
disji.1 (𝑥 = 𝑋𝐵 = 𝐶)
disji.2 (𝑥 = 𝑌𝐵 = 𝐷)
Assertion
Ref Expression
disji ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ (𝑍𝐶𝑍𝐷)) → 𝑋 = 𝑌)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝑋   𝑥,𝑌
Allowed substitution hints:   𝐵(𝑥)   𝑍(𝑥)

Proof of Theorem disji
StepHypRef Expression
1 inelcm 3984 . 2 ((𝑍𝐶𝑍𝐷) → (𝐶𝐷) ≠ ∅)
2 disji.1 . . . . . 6 (𝑥 = 𝑋𝐵 = 𝐶)
3 disji.2 . . . . . 6 (𝑥 = 𝑌𝐵 = 𝐷)
42, 3disji2 4569 . . . . 5 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ 𝑋𝑌) → (𝐶𝐷) = ∅)
543expia 1259 . . . 4 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴)) → (𝑋𝑌 → (𝐶𝐷) = ∅))
65necon1d 2804 . . 3 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴)) → ((𝐶𝐷) ≠ ∅ → 𝑋 = 𝑌))
763impia 1253 . 2 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ (𝐶𝐷) ≠ ∅) → 𝑋 = 𝑌)
81, 7syl3an3 1353 1 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ (𝑍𝐶𝑍𝐷)) → 𝑋 = 𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ 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:  volfiniun  23122
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