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Theorem disji2f 32242
Description: Property of a disjoint collection: if 𝐵(𝑥) = 𝐶 and 𝐵(𝑌) = 𝐷, and 𝑥𝑌, then 𝐵 and 𝐶 are disjoint. (Contributed by Thierry Arnoux, 30-Dec-2016.)
Hypotheses
Ref Expression
disjif.1 𝑥𝐶
disjif.2 (𝑥 = 𝑌𝐵 = 𝐶)
Assertion
Ref Expression
disji2f ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ 𝑥𝑌) → (𝐵𝐶) = ∅)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑌
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem disji2f
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ne 2940 . . 3 (𝑥𝑌 ↔ ¬ 𝑥 = 𝑌)
2 disjors 5129 . . . . . 6 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅))
3 equequ1 2027 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦 = 𝑧𝑥 = 𝑧))
4 csbeq1 3896 . . . . . . . . . . 11 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
5 csbid 3906 . . . . . . . . . . 11 𝑥 / 𝑥𝐵 = 𝐵
64, 5eqtrdi 2787 . . . . . . . . . 10 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
76ineq1d 4211 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = (𝐵𝑧 / 𝑥𝐵))
87eqeq1d 2733 . . . . . . . 8 (𝑦 = 𝑥 → ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ ↔ (𝐵𝑧 / 𝑥𝐵) = ∅))
93, 8orbi12d 916 . . . . . . 7 (𝑦 = 𝑥 → ((𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) ↔ (𝑥 = 𝑧 ∨ (𝐵𝑧 / 𝑥𝐵) = ∅)))
10 eqeq2 2743 . . . . . . . 8 (𝑧 = 𝑌 → (𝑥 = 𝑧𝑥 = 𝑌))
11 nfcv 2902 . . . . . . . . . . 11 𝑥𝑌
12 disjif.1 . . . . . . . . . . 11 𝑥𝐶
13 disjif.2 . . . . . . . . . . 11 (𝑥 = 𝑌𝐵 = 𝐶)
1411, 12, 13csbhypf 3922 . . . . . . . . . 10 (𝑧 = 𝑌𝑧 / 𝑥𝐵 = 𝐶)
1514ineq2d 4212 . . . . . . . . 9 (𝑧 = 𝑌 → (𝐵𝑧 / 𝑥𝐵) = (𝐵𝐶))
1615eqeq1d 2733 . . . . . . . 8 (𝑧 = 𝑌 → ((𝐵𝑧 / 𝑥𝐵) = ∅ ↔ (𝐵𝐶) = ∅))
1710, 16orbi12d 916 . . . . . . 7 (𝑧 = 𝑌 → ((𝑥 = 𝑧 ∨ (𝐵𝑧 / 𝑥𝐵) = ∅) ↔ (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅)))
189, 17rspc2v 3622 . . . . . 6 ((𝑥𝐴𝑌𝐴) → (∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅)))
192, 18biimtrid 241 . . . . 5 ((𝑥𝐴𝑌𝐴) → (Disj 𝑥𝐴 𝐵 → (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅)))
2019impcom 407 . . . 4 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅))
2120ord 861 . . 3 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → (¬ 𝑥 = 𝑌 → (𝐵𝐶) = ∅))
221, 21biimtrid 241 . 2 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → (𝑥𝑌 → (𝐵𝐶) = ∅))
23223impia 1116 1 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ 𝑥𝑌) → (𝐵𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 844  w3a 1086   = wceq 1540  wcel 2105  wnfc 2882  wne 2939  wral 3060  csb 3893  cin 3947  c0 4322  Disj wdisj 5113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rmo 3375  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-in 3955  df-nul 4323  df-disj 5114
This theorem is referenced by:  disjif  32243
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