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Theorem disjif2 30325
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 4413 . 2 ((𝑍𝐵𝑍𝐶) → (𝐵𝐶) ≠ ∅)
2 disjif2.1 . . . . . . . 8 𝑥𝐴
32disjorsf 30324 . . . . . . 7 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅))
4 equequ1 2028 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦 = 𝑧𝑥 = 𝑧))
5 csbeq1 3885 . . . . . . . . . . . 12 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
6 csbid 3895 . . . . . . . . . . . 12 𝑥 / 𝑥𝐵 = 𝐵
75, 6syl6eq 2872 . . . . . . . . . . 11 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
87ineq1d 4187 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = (𝐵𝑧 / 𝑥𝐵))
98eqeq1d 2823 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ ↔ (𝐵𝑧 / 𝑥𝐵) = ∅))
104, 9orbi12d 915 . . . . . . . 8 (𝑦 = 𝑥 → ((𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) ↔ (𝑥 = 𝑧 ∨ (𝐵𝑧 / 𝑥𝐵) = ∅)))
11 eqeq2 2833 . . . . . . . . 9 (𝑧 = 𝑌 → (𝑥 = 𝑧𝑥 = 𝑌))
12 nfcv 2977 . . . . . . . . . . . 12 𝑥𝑌
13 disjif2.2 . . . . . . . . . . . 12 𝑥𝐶
14 disjif2.3 . . . . . . . . . . . 12 (𝑥 = 𝑌𝐵 = 𝐶)
1512, 13, 14csbhypf 3910 . . . . . . . . . . 11 (𝑧 = 𝑌𝑧 / 𝑥𝐵 = 𝐶)
1615ineq2d 4188 . . . . . . . . . 10 (𝑧 = 𝑌 → (𝐵𝑧 / 𝑥𝐵) = (𝐵𝐶))
1716eqeq1d 2823 . . . . . . . . 9 (𝑧 = 𝑌 → ((𝐵𝑧 / 𝑥𝐵) = ∅ ↔ (𝐵𝐶) = ∅))
1811, 17orbi12d 915 . . . . . . . 8 (𝑧 = 𝑌 → ((𝑥 = 𝑧 ∨ (𝐵𝑧 / 𝑥𝐵) = ∅) ↔ (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅)))
1910, 18rspc2v 3632 . . . . . . 7 ((𝑥𝐴𝑌𝐴) → (∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅)))
203, 19syl5bi 244 . . . . . 6 ((𝑥𝐴𝑌𝐴) → (Disj 𝑥𝐴 𝐵 → (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅)))
2120impcom 410 . . . . 5 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅))
2221ord 860 . . . 4 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → (¬ 𝑥 = 𝑌 → (𝐵𝐶) = ∅))
2322necon1ad 3033 . . 3 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → ((𝐵𝐶) ≠ ∅ → 𝑥 = 𝑌))
24233impia 1113 . 2 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝐵𝐶) ≠ ∅) → 𝑥 = 𝑌)
251, 24syl3an3 1161 1 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝑍𝐵𝑍𝐶)) → 𝑥 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wo 843  w3a 1083   = wceq 1533  wcel 2110  wnfc 2961  wne 3016  wral 3138  csb 3882  cin 3934  c0 4290  Disj wdisj 5023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-in 3942  df-nul 4291  df-disj 5024
This theorem is referenced by:  disjabrexf  30327
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