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Theorem disji2 4563
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 2781 . . 3 (𝑋𝑌 ↔ ¬ 𝑋 = 𝑌)
2 disjors 4562 . . . . . 6 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅))
3 eqeq1 2613 . . . . . . . 8 (𝑦 = 𝑋 → (𝑦 = 𝑧𝑋 = 𝑧))
4 nfcv 2750 . . . . . . . . . . 11 𝑥𝑋
5 nfcv 2750 . . . . . . . . . . 11 𝑥𝐶
6 disji.1 . . . . . . . . . . 11 (𝑥 = 𝑋𝐵 = 𝐶)
74, 5, 6csbhypf 3517 . . . . . . . . . 10 (𝑦 = 𝑋𝑦 / 𝑥𝐵 = 𝐶)
87ineq1d 3774 . . . . . . . . 9 (𝑦 = 𝑋 → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = (𝐶𝑧 / 𝑥𝐵))
98eqeq1d 2611 . . . . . . . 8 (𝑦 = 𝑋 → ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ ↔ (𝐶𝑧 / 𝑥𝐵) = ∅))
103, 9orbi12d 741 . . . . . . 7 (𝑦 = 𝑋 → ((𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) ↔ (𝑋 = 𝑧 ∨ (𝐶𝑧 / 𝑥𝐵) = ∅)))
11 eqeq2 2620 . . . . . . . 8 (𝑧 = 𝑌 → (𝑋 = 𝑧𝑋 = 𝑌))
12 nfcv 2750 . . . . . . . . . . 11 𝑥𝑌
13 nfcv 2750 . . . . . . . . . . 11 𝑥𝐷
14 disji.2 . . . . . . . . . . 11 (𝑥 = 𝑌𝐵 = 𝐷)
1512, 13, 14csbhypf 3517 . . . . . . . . . 10 (𝑧 = 𝑌𝑧 / 𝑥𝐵 = 𝐷)
1615ineq2d 3775 . . . . . . . . 9 (𝑧 = 𝑌 → (𝐶𝑧 / 𝑥𝐵) = (𝐶𝐷))
1716eqeq1d 2611 . . . . . . . 8 (𝑧 = 𝑌 → ((𝐶𝑧 / 𝑥𝐵) = ∅ ↔ (𝐶𝐷) = ∅))
1811, 17orbi12d 741 . . . . . . 7 (𝑧 = 𝑌 → ((𝑋 = 𝑧 ∨ (𝐶𝑧 / 𝑥𝐵) = ∅) ↔ (𝑋 = 𝑌 ∨ (𝐶𝐷) = ∅)))
1910, 18rspc2v 3292 . . . . . 6 ((𝑋𝐴𝑌𝐴) → (∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → (𝑋 = 𝑌 ∨ (𝐶𝐷) = ∅)))
202, 19syl5bi 230 . . . . 5 ((𝑋𝐴𝑌𝐴) → (Disj 𝑥𝐴 𝐵 → (𝑋 = 𝑌 ∨ (𝐶𝐷) = ∅)))
2120impcom 444 . . . 4 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴)) → (𝑋 = 𝑌 ∨ (𝐶𝐷) = ∅))
2221ord 390 . . 3 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴)) → (¬ 𝑋 = 𝑌 → (𝐶𝐷) = ∅))
231, 22syl5bi 230 . 2 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴)) → (𝑋𝑌 → (𝐶𝐷) = ∅))
24233impia 1252 1 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ 𝑋𝑌) → (𝐶𝐷) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  csb 3498  cin 3538  c0 3873  Disj wdisj 4547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-in 3546  df-nul 3874  df-disj 4548
This theorem is referenced by:  disji  4564  disjxiun  4573  disjxiunOLD  4574  voliunlem1  23042  disjf1  38167
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