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Theorem disjprg 4572
Description: A pair collection is disjoint iff the two sets in the family have empty intersection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
disjprg.1 (𝑥 = 𝐴𝐶 = 𝐷)
disjprg.2 (𝑥 = 𝐵𝐶 = 𝐸)
Assertion
Ref Expression
disjprg ((𝐴𝑉𝐵𝑉𝐴𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ (𝐷𝐸) = ∅))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝐸
Allowed substitution hints:   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem disjprg
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2613 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 = 𝑧𝐴 = 𝑧))
2 nfcv 2750 . . . . . . . . . 10 𝑥𝐴
3 nfcv 2750 . . . . . . . . . 10 𝑥𝐷
4 disjprg.1 . . . . . . . . . 10 (𝑥 = 𝐴𝐶 = 𝐷)
52, 3, 4csbhypf 3517 . . . . . . . . 9 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐷)
65ineq1d 3774 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = (𝐷𝑧 / 𝑥𝐶))
76eqeq1d 2611 . . . . . . 7 (𝑦 = 𝐴 → ((𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅ ↔ (𝐷𝑧 / 𝑥𝐶) = ∅))
81, 7orbi12d 741 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅) ↔ (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅)))
98ralbidv 2968 . . . . 5 (𝑦 = 𝐴 → (∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅) ↔ ∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅)))
10 eqeq1 2613 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 = 𝑧𝐵 = 𝑧))
11 nfcv 2750 . . . . . . . . . 10 𝑥𝐵
12 nfcv 2750 . . . . . . . . . 10 𝑥𝐸
13 disjprg.2 . . . . . . . . . 10 (𝑥 = 𝐵𝐶 = 𝐸)
1411, 12, 13csbhypf 3517 . . . . . . . . 9 (𝑦 = 𝐵𝑦 / 𝑥𝐶 = 𝐸)
1514ineq1d 3774 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = (𝐸𝑧 / 𝑥𝐶))
1615eqeq1d 2611 . . . . . . 7 (𝑦 = 𝐵 → ((𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅ ↔ (𝐸𝑧 / 𝑥𝐶) = ∅))
1710, 16orbi12d 741 . . . . . 6 (𝑦 = 𝐵 → ((𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅) ↔ (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅)))
1817ralbidv 2968 . . . . 5 (𝑦 = 𝐵 → (∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅) ↔ ∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅)))
199, 18ralprg 4180 . . . 4 ((𝐴𝑉𝐵𝑉) → (∀𝑦 ∈ {𝐴, 𝐵}∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅) ↔ (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ∧ ∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅))))
20193adant3 1073 . . 3 ((𝐴𝑉𝐵𝑉𝐴𝐵) → (∀𝑦 ∈ {𝐴, 𝐵}∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅) ↔ (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ∧ ∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅))))
21 id 22 . . . . . . . . . 10 (𝑧 = 𝐴𝑧 = 𝐴)
2221eqcomd 2615 . . . . . . . . 9 (𝑧 = 𝐴𝐴 = 𝑧)
2322orcd 405 . . . . . . . 8 (𝑧 = 𝐴 → (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅))
24 a1tru 1490 . . . . . . . 8 (𝑧 = 𝐴 → ⊤)
2523, 242thd 253 . . . . . . 7 (𝑧 = 𝐴 → ((𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ↔ ⊤))
26 eqeq2 2620 . . . . . . . 8 (𝑧 = 𝐵 → (𝐴 = 𝑧𝐴 = 𝐵))
2711, 12, 13csbhypf 3517 . . . . . . . . . 10 (𝑧 = 𝐵𝑧 / 𝑥𝐶 = 𝐸)
2827ineq2d 3775 . . . . . . . . 9 (𝑧 = 𝐵 → (𝐷𝑧 / 𝑥𝐶) = (𝐷𝐸))
2928eqeq1d 2611 . . . . . . . 8 (𝑧 = 𝐵 → ((𝐷𝑧 / 𝑥𝐶) = ∅ ↔ (𝐷𝐸) = ∅))
3026, 29orbi12d 741 . . . . . . 7 (𝑧 = 𝐵 → ((𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ↔ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅)))
3125, 30ralprg 4180 . . . . . 6 ((𝐴𝑉𝐵𝑉) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅))))
32313adant3 1073 . . . . 5 ((𝐴𝑉𝐵𝑉𝐴𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅))))
33 simp3 1055 . . . . . . . 8 ((𝐴𝑉𝐵𝑉𝐴𝐵) → 𝐴𝐵)
3433neneqd 2786 . . . . . . 7 ((𝐴𝑉𝐵𝑉𝐴𝐵) → ¬ 𝐴 = 𝐵)
35 biorf 418 . . . . . . 7 𝐴 = 𝐵 → ((𝐷𝐸) = ∅ ↔ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅)))
3634, 35syl 17 . . . . . 6 ((𝐴𝑉𝐵𝑉𝐴𝐵) → ((𝐷𝐸) = ∅ ↔ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅)))
37 tru 1478 . . . . . . 7
3837biantrur 525 . . . . . 6 ((𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅) ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅)))
3936, 38syl6bb 274 . . . . 5 ((𝐴𝑉𝐵𝑉𝐴𝐵) → ((𝐷𝐸) = ∅ ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅))))
4032, 39bitr4d 269 . . . 4 ((𝐴𝑉𝐵𝑉𝐴𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ↔ (𝐷𝐸) = ∅))
41 eqeq2 2620 . . . . . . . . 9 (𝑧 = 𝐴 → (𝐵 = 𝑧𝐵 = 𝐴))
42 eqcom 2616 . . . . . . . . 9 (𝐵 = 𝐴𝐴 = 𝐵)
4341, 42syl6bb 274 . . . . . . . 8 (𝑧 = 𝐴 → (𝐵 = 𝑧𝐴 = 𝐵))
442, 3, 4csbhypf 3517 . . . . . . . . . . 11 (𝑧 = 𝐴𝑧 / 𝑥𝐶 = 𝐷)
4544ineq2d 3775 . . . . . . . . . 10 (𝑧 = 𝐴 → (𝐸𝑧 / 𝑥𝐶) = (𝐸𝐷))
46 incom 3766 . . . . . . . . . 10 (𝐸𝐷) = (𝐷𝐸)
4745, 46syl6eq 2659 . . . . . . . . 9 (𝑧 = 𝐴 → (𝐸𝑧 / 𝑥𝐶) = (𝐷𝐸))
4847eqeq1d 2611 . . . . . . . 8 (𝑧 = 𝐴 → ((𝐸𝑧 / 𝑥𝐶) = ∅ ↔ (𝐷𝐸) = ∅))
4943, 48orbi12d 741 . . . . . . 7 (𝑧 = 𝐴 → ((𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅) ↔ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅)))
50 id 22 . . . . . . . . . 10 (𝑧 = 𝐵𝑧 = 𝐵)
5150eqcomd 2615 . . . . . . . . 9 (𝑧 = 𝐵𝐵 = 𝑧)
5251orcd 405 . . . . . . . 8 (𝑧 = 𝐵 → (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅))
53 a1tru 1490 . . . . . . . 8 (𝑧 = 𝐵 → ⊤)
5452, 532thd 253 . . . . . . 7 (𝑧 = 𝐵 → ((𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅) ↔ ⊤))
5549, 54ralprg 4180 . . . . . 6 ((𝐴𝑉𝐵𝑉) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅) ↔ ((𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅) ∧ ⊤)))
56553adant3 1073 . . . . 5 ((𝐴𝑉𝐵𝑉𝐴𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅) ↔ ((𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅) ∧ ⊤)))
5737biantru 524 . . . . . 6 ((𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅) ↔ ((𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅) ∧ ⊤))
5836, 57syl6bb 274 . . . . 5 ((𝐴𝑉𝐵𝑉𝐴𝐵) → ((𝐷𝐸) = ∅ ↔ ((𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅) ∧ ⊤)))
5956, 58bitr4d 269 . . . 4 ((𝐴𝑉𝐵𝑉𝐴𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅) ↔ (𝐷𝐸) = ∅))
6040, 59anbi12d 742 . . 3 ((𝐴𝑉𝐵𝑉𝐴𝐵) → ((∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ∧ ∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅)) ↔ ((𝐷𝐸) = ∅ ∧ (𝐷𝐸) = ∅)))
6120, 60bitrd 266 . 2 ((𝐴𝑉𝐵𝑉𝐴𝐵) → (∀𝑦 ∈ {𝐴, 𝐵}∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅) ↔ ((𝐷𝐸) = ∅ ∧ (𝐷𝐸) = ∅)))
62 disjors 4562 . 2 (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ ∀𝑦 ∈ {𝐴, 𝐵}∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅))
63 pm4.24 672 . 2 ((𝐷𝐸) = ∅ ↔ ((𝐷𝐸) = ∅ ∧ (𝐷𝐸) = ∅))
6461, 62, 633bitr4g 301 1 ((𝐴𝑉𝐵𝑉𝐴𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ (𝐷𝐸) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wtru 1475  wcel 1976  wne 2779  wral 2895  csb 3498  cin 3538  c0 3873  {cpr 4126  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-un 3544  df-in 3546  df-nul 3874  df-sn 4125  df-pr 4127  df-disj 4548
This theorem is referenced by:  disjdifprg  28576  unelldsys  29354  pmeasmono  29519  probun  29614  meadjun  39159
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