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Theorem disji2f 29697
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 2933 . . 3 (𝑥𝑌 ↔ ¬ 𝑥 = 𝑌)
2 disjors 4787 . . . . . 6 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅))
3 equequ1 2107 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦 = 𝑧𝑥 = 𝑧))
4 csbeq1 3677 . . . . . . . . . . 11 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
5 csbid 3682 . . . . . . . . . . 11 𝑥 / 𝑥𝐵 = 𝐵
64, 5syl6eq 2810 . . . . . . . . . 10 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
76ineq1d 3956 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = (𝐵𝑧 / 𝑥𝐵))
87eqeq1d 2762 . . . . . . . 8 (𝑦 = 𝑥 → ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ ↔ (𝐵𝑧 / 𝑥𝐵) = ∅))
93, 8orbi12d 748 . . . . . . 7 (𝑦 = 𝑥 → ((𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) ↔ (𝑥 = 𝑧 ∨ (𝐵𝑧 / 𝑥𝐵) = ∅)))
10 eqeq2 2771 . . . . . . . 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 3693 . . . . . . . . . 10 (𝑧 = 𝑌𝑧 / 𝑥𝐵 = 𝐶)
1514ineq2d 3957 . . . . . . . . 9 (𝑧 = 𝑌 → (𝐵𝑧 / 𝑥𝐵) = (𝐵𝐶))
1615eqeq1d 2762 . . . . . . . 8 (𝑧 = 𝑌 → ((𝐵𝑧 / 𝑥𝐵) = ∅ ↔ (𝐵𝐶) = ∅))
1710, 16orbi12d 748 . . . . . . 7 (𝑧 = 𝑌 → ((𝑥 = 𝑧 ∨ (𝐵𝑧 / 𝑥𝐵) = ∅) ↔ (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅)))
189, 17rspc2v 3461 . . . . . 6 ((𝑥𝐴𝑌𝐴) → (∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅)))
192, 18syl5bi 232 . . . . 5 ((𝑥𝐴𝑌𝐴) → (Disj 𝑥𝐴 𝐵 → (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅)))
2019impcom 445 . . . 4 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → (𝑥 = 𝑌 ∨ (𝐵𝐶) = ∅))
2120ord 391 . . 3 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → (¬ 𝑥 = 𝑌 → (𝐵𝐶) = ∅))
221, 21syl5bi 232 . 2 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴)) → (𝑥𝑌 → (𝐵𝐶) = ∅))
23223impia 1110 1 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ 𝑥𝑌) → (𝐵𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  wnfc 2889  wne 2932  wral 3050  csb 3674  cin 3714  c0 4058  Disj wdisj 4772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-in 3722  df-nul 4059  df-disj 4773
This theorem is referenced by:  disjif  29698
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