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Theorem disjif2 29257
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, 6-Apr-2017.)
Hypotheses
Ref Expression
disjif2.1 𝑥𝐴
disjif2.2 𝑥𝐶
disjif2.3 (𝑥 = 𝑌𝐵 = 𝐶)
Assertion
Ref Expression
disjif2 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝑍𝐵𝑍𝐶)) → 𝑥 = 𝑌)
Distinct variable group:   𝑥,𝑌
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝑍(𝑥)

Proof of Theorem disjif2
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inelcm 4009 . 2 ((𝑍𝐵𝑍𝐶) → (𝐵𝐶) ≠ ∅)
2 disjif2.1 . . . . . . . 8 𝑥𝐴
32disjorsf 29256 . . . . . . 7 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅))
4 equequ1 1949 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦 = 𝑧𝑥 = 𝑧))
5 csbeq1 3521 . . . . . . . . . . . 12 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
6 csbid 3526 . . . . . . . . . . . 12 𝑥 / 𝑥𝐵 = 𝐵
75, 6syl6eq 2671 . . . . . . . . . . 11 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
87ineq1d 3796 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = (𝐵𝑧 / 𝑥𝐵))
98eqeq1d 2623 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ ↔ (𝐵𝑧 / 𝑥𝐵) = ∅))
104, 9orbi12d 745 . . . . . . . 8 (𝑦 = 𝑥 → ((𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) ↔ (𝑥 = 𝑧 ∨ (𝐵𝑧 / 𝑥𝐵) = ∅)))
11 eqeq2 2632 . . . . . . . . 9 (𝑧 = 𝑌 → (𝑥 = 𝑧𝑥 = 𝑌))
12 nfcv 2761 . . . . . . . . . . . 12 𝑥𝑌
13 disjif2.2 . . . . . . . . . . . 12 𝑥𝐶
14 disjif2.3 . . . . . . . . . . . 12 (𝑥 = 𝑌𝐵 = 𝐶)
1512, 13, 14csbhypf 3537 . . . . . . . . . . 11 (𝑧 = 𝑌𝑧 / 𝑥𝐵 = 𝐶)
1615ineq2d 3797 . . . . . . . . . 10 (𝑧 = 𝑌 → (𝐵𝑧 / 𝑥𝐵) = (𝐵𝐶))
1716eqeq1d 2623 . . . . . . . . 9 (𝑧 = 𝑌 → ((𝐵𝑧 / 𝑥𝐵) = ∅ ↔ (𝐵𝐶) = ∅))
1811, 17orbi12d 745 . . . . . . . 8 (𝑧 = 𝑌 → ((𝑥 = 𝑧 ∨ (𝐵𝑧 / 𝑥𝐵) = ∅) ↔ (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅)))
1910, 18rspc2v 3310 . . . . . . 7 ((𝑥𝐴𝑌𝐴) → (∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅)))
203, 19syl5bi 232 . . . . . 6 ((𝑥𝐴𝑌𝐴) → (Disj 𝑥𝐴 𝐵 → (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅)))
2120impcom 446 . . . . 5 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅))
2221ord 392 . . . 4 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → (¬ 𝑥 = 𝑌 → (𝐵𝐶) = ∅))
2322necon1ad 2807 . . 3 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → ((𝐵𝐶) ≠ ∅ → 𝑥 = 𝑌))
24233impia 1258 . 2 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝐵𝐶) ≠ ∅) → 𝑥 = 𝑌)
251, 24syl3an3 1358 1 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝑍𝐵𝑍𝐶)) → 𝑥 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wnfc 2748  wne 2790  wral 2907  csb 3518  cin 3558  c0 3896  Disj wdisj 4588
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 2912  df-rmo 2915  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-in 3566  df-nul 3897  df-disj 4589
This theorem is referenced by:  disjabrexf  29259
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