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Theorem disji2 4788
Description: Property of a disjoint collection: if 𝐵(𝑋) = 𝐶 and 𝐵(𝑌) = 𝐷, and 𝑋𝑌, then 𝐶 and 𝐷 are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
disji.1 (𝑥 = 𝑋𝐵 = 𝐶)
disji.2 (𝑥 = 𝑌𝐵 = 𝐷)
Assertion
Ref Expression
disji2 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ 𝑋𝑌) → (𝐶𝐷) = ∅)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝑋   𝑥,𝑌
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem disji2
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 eqeq1 2764 . . . . . . . 8 (𝑦 = 𝑋 → (𝑦 = 𝑧𝑋 = 𝑧))
4 nfcv 2902 . . . . . . . . . . 11 𝑥𝑋
5 nfcv 2902 . . . . . . . . . . 11 𝑥𝐶
6 disji.1 . . . . . . . . . . 11 (𝑥 = 𝑋𝐵 = 𝐶)
74, 5, 6csbhypf 3693 . . . . . . . . . 10 (𝑦 = 𝑋𝑦 / 𝑥𝐵 = 𝐶)
87ineq1d 3956 . . . . . . . . 9 (𝑦 = 𝑋 → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = (𝐶𝑧 / 𝑥𝐵))
98eqeq1d 2762 . . . . . . . 8 (𝑦 = 𝑋 → ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ ↔ (𝐶𝑧 / 𝑥𝐵) = ∅))
103, 9orbi12d 748 . . . . . . 7 (𝑦 = 𝑋 → ((𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) ↔ (𝑋 = 𝑧 ∨ (𝐶𝑧 / 𝑥𝐵) = ∅)))
11 eqeq2 2771 . . . . . . . 8 (𝑧 = 𝑌 → (𝑋 = 𝑧𝑋 = 𝑌))
12 nfcv 2902 . . . . . . . . . . 11 𝑥𝑌
13 nfcv 2902 . . . . . . . . . . 11 𝑥𝐷
14 disji.2 . . . . . . . . . . 11 (𝑥 = 𝑌𝐵 = 𝐷)
1512, 13, 14csbhypf 3693 . . . . . . . . . 10 (𝑧 = 𝑌𝑧 / 𝑥𝐵 = 𝐷)
1615ineq2d 3957 . . . . . . . . 9 (𝑧 = 𝑌 → (𝐶𝑧 / 𝑥𝐵) = (𝐶𝐷))
1716eqeq1d 2762 . . . . . . . 8 (𝑧 = 𝑌 → ((𝐶𝑧 / 𝑥𝐵) = ∅ ↔ (𝐶𝐷) = ∅))
1811, 17orbi12d 748 . . . . . . 7 (𝑧 = 𝑌 → ((𝑋 = 𝑧 ∨ (𝐶𝑧 / 𝑥𝐵) = ∅) ↔ (𝑋 = 𝑌 ∨ (𝐶𝐷) = ∅)))
1910, 18rspc2v 3461 . . . . . 6 ((𝑋𝐴𝑌𝐴) → (∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → (𝑋 = 𝑌 ∨ (𝐶𝐷) = ∅)))
202, 19syl5bi 232 . . . . 5 ((𝑋𝐴𝑌𝐴) → (Disj 𝑥𝐴 𝐵 → (𝑋 = 𝑌 ∨ (𝐶𝐷) = ∅)))
2120impcom 445 . . . 4 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴)) → (𝑋 = 𝑌 ∨ (𝐶𝐷) = ∅))
2221ord 391 . . 3 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴)) → (¬ 𝑋 = 𝑌 → (𝐶𝐷) = ∅))
231, 22syl5bi 232 . 2 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴)) → (𝑋𝑌 → (𝐶𝐷) = ∅))
24233impia 1110 1 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ 𝑋𝑌) → (𝐶𝐷) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  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:  disji  4789  disjxiun  4801  voliunlem1  23518  disjf1  39868
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