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Theorem disji2f 29773
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 2938 . . 3 (𝑥𝑌 ↔ ¬ 𝑥 = 𝑌)
2 disjors 4792 . . . . . 6 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅))
3 equequ1 2122 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦 = 𝑧𝑥 = 𝑧))
4 csbeq1 3694 . . . . . . . . . . 11 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
5 csbid 3699 . . . . . . . . . . 11 𝑥 / 𝑥𝐵 = 𝐵
64, 5syl6eq 2815 . . . . . . . . . 10 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
76ineq1d 3975 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = (𝐵𝑧 / 𝑥𝐵))
87eqeq1d 2767 . . . . . . . 8 (𝑦 = 𝑥 → ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ ↔ (𝐵𝑧 / 𝑥𝐵) = ∅))
93, 8orbi12d 942 . . . . . . 7 (𝑦 = 𝑥 → ((𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) ↔ (𝑥 = 𝑧 ∨ (𝐵𝑧 / 𝑥𝐵) = ∅)))
10 eqeq2 2776 . . . . . . . 8 (𝑧 = 𝑌 → (𝑥 = 𝑧𝑥 = 𝑌))
11 nfcv 2907 . . . . . . . . . . 11 𝑥𝑌
12 disjif.1 . . . . . . . . . . 11 𝑥𝐶
13 disjif.2 . . . . . . . . . . 11 (𝑥 = 𝑌𝐵 = 𝐶)
1411, 12, 13csbhypf 3710 . . . . . . . . . 10 (𝑧 = 𝑌𝑧 / 𝑥𝐵 = 𝐶)
1514ineq2d 3976 . . . . . . . . 9 (𝑧 = 𝑌 → (𝐵𝑧 / 𝑥𝐵) = (𝐵𝐶))
1615eqeq1d 2767 . . . . . . . 8 (𝑧 = 𝑌 → ((𝐵𝑧 / 𝑥𝐵) = ∅ ↔ (𝐵𝐶) = ∅))
1710, 16orbi12d 942 . . . . . . 7 (𝑧 = 𝑌 → ((𝑥 = 𝑧 ∨ (𝐵𝑧 / 𝑥𝐵) = ∅) ↔ (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅)))
189, 17rspc2v 3474 . . . . . 6 ((𝑥𝐴𝑌𝐴) → (∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅)))
192, 18syl5bi 233 . . . . 5 ((𝑥𝐴𝑌𝐴) → (Disj 𝑥𝐴 𝐵 → (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅)))
2019impcom 396 . . . 4 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅))
2120ord 890 . . 3 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → (¬ 𝑥 = 𝑌 → (𝐵𝐶) = ∅))
221, 21syl5bi 233 . 2 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → (𝑥𝑌 → (𝐵𝐶) = ∅))
23223impia 1145 1 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ 𝑥𝑌) → (𝐵𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wnfc 2894  wne 2937  wral 3055  csb 3691  cin 3731  c0 4079  Disj wdisj 4777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-in 3739  df-nul 4080  df-disj 4778
This theorem is referenced by:  disjif  29774
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