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Theorem disjprg 5053
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 2822 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 = 𝑧𝐴 = 𝑧))
2 nfcv 2974 . . . . . . . . . 10 𝑥𝐴
3 nfcv 2974 . . . . . . . . . 10 𝑥𝐷
4 disjprg.1 . . . . . . . . . 10 (𝑥 = 𝐴𝐶 = 𝐷)
52, 3, 4csbhypf 3908 . . . . . . . . 9 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐷)
65ineq1d 4185 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = (𝐷𝑧 / 𝑥𝐶))
76eqeq1d 2820 . . . . . . 7 (𝑦 = 𝐴 → ((𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅ ↔ (𝐷𝑧 / 𝑥𝐶) = ∅))
81, 7orbi12d 912 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅) ↔ (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅)))
98ralbidv 3194 . . . . 5 (𝑦 = 𝐴 → (∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅) ↔ ∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅)))
10 eqeq1 2822 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 = 𝑧𝐵 = 𝑧))
11 nfcv 2974 . . . . . . . . . 10 𝑥𝐵
12 nfcv 2974 . . . . . . . . . 10 𝑥𝐸
13 disjprg.2 . . . . . . . . . 10 (𝑥 = 𝐵𝐶 = 𝐸)
1411, 12, 13csbhypf 3908 . . . . . . . . 9 (𝑦 = 𝐵𝑦 / 𝑥𝐶 = 𝐸)
1514ineq1d 4185 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = (𝐸𝑧 / 𝑥𝐶))
1615eqeq1d 2820 . . . . . . 7 (𝑦 = 𝐵 → ((𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅ ↔ (𝐸𝑧 / 𝑥𝐶) = ∅))
1710, 16orbi12d 912 . . . . . 6 (𝑦 = 𝐵 → ((𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅) ↔ (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅)))
1817ralbidv 3194 . . . . 5 (𝑦 = 𝐵 → (∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅) ↔ ∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅)))
199, 18ralprg 4624 . . . 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 1124 . . 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 2824 . . . . . . . . 9 (𝑧 = 𝐴𝐴 = 𝑧)
2322orcd 869 . . . . . . . 8 (𝑧 = 𝐴 → (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅))
24 trud 1538 . . . . . . . 8 (𝑧 = 𝐴 → ⊤)
2523, 242thd 266 . . . . . . 7 (𝑧 = 𝐴 → ((𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ↔ ⊤))
26 eqeq2 2830 . . . . . . . 8 (𝑧 = 𝐵 → (𝐴 = 𝑧𝐴 = 𝐵))
2711, 12, 13csbhypf 3908 . . . . . . . . . 10 (𝑧 = 𝐵𝑧 / 𝑥𝐶 = 𝐸)
2827ineq2d 4186 . . . . . . . . 9 (𝑧 = 𝐵 → (𝐷𝑧 / 𝑥𝐶) = (𝐷𝐸))
2928eqeq1d 2820 . . . . . . . 8 (𝑧 = 𝐵 → ((𝐷𝑧 / 𝑥𝐶) = ∅ ↔ (𝐷𝐸) = ∅))
3026, 29orbi12d 912 . . . . . . 7 (𝑧 = 𝐵 → ((𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ↔ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅)))
3125, 30ralprg 4624 . . . . . 6 ((𝐴𝑉𝐵𝑉) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅))))
32313adant3 1124 . . . . 5 ((𝐴𝑉𝐵𝑉𝐴𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅))))
33 simp3 1130 . . . . . . . 8 ((𝐴𝑉𝐵𝑉𝐴𝐵) → 𝐴𝐵)
3433neneqd 3018 . . . . . . 7 ((𝐴𝑉𝐵𝑉𝐴𝐵) → ¬ 𝐴 = 𝐵)
35 biorf 930 . . . . . . 7 𝐴 = 𝐵 → ((𝐷𝐸) = ∅ ↔ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅)))
3634, 35syl 17 . . . . . 6 ((𝐴𝑉𝐵𝑉𝐴𝐵) → ((𝐷𝐸) = ∅ ↔ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅)))
37 tru 1532 . . . . . . 7
3837biantrur 531 . . . . . 6 ((𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅) ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅)))
3936, 38syl6bb 288 . . . . 5 ((𝐴𝑉𝐵𝑉𝐴𝐵) → ((𝐷𝐸) = ∅ ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅))))
4032, 39bitr4d 283 . . . 4 ((𝐴𝑉𝐵𝑉𝐴𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ↔ (𝐷𝐸) = ∅))
41 eqeq2 2830 . . . . . . . . 9 (𝑧 = 𝐴 → (𝐵 = 𝑧𝐵 = 𝐴))
42 eqcom 2825 . . . . . . . . 9 (𝐵 = 𝐴𝐴 = 𝐵)
4341, 42syl6bb 288 . . . . . . . 8 (𝑧 = 𝐴 → (𝐵 = 𝑧𝐴 = 𝐵))
442, 3, 4csbhypf 3908 . . . . . . . . . . 11 (𝑧 = 𝐴𝑧 / 𝑥𝐶 = 𝐷)
4544ineq2d 4186 . . . . . . . . . 10 (𝑧 = 𝐴 → (𝐸𝑧 / 𝑥𝐶) = (𝐸𝐷))
46 incom 4175 . . . . . . . . . 10 (𝐸𝐷) = (𝐷𝐸)
4745, 46syl6eq 2869 . . . . . . . . 9 (𝑧 = 𝐴 → (𝐸𝑧 / 𝑥𝐶) = (𝐷𝐸))
4847eqeq1d 2820 . . . . . . . 8 (𝑧 = 𝐴 → ((𝐸𝑧 / 𝑥𝐶) = ∅ ↔ (𝐷𝐸) = ∅))
4943, 48orbi12d 912 . . . . . . 7 (𝑧 = 𝐴 → ((𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅) ↔ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅)))
50 id 22 . . . . . . . . . 10 (𝑧 = 𝐵𝑧 = 𝐵)
5150eqcomd 2824 . . . . . . . . 9 (𝑧 = 𝐵𝐵 = 𝑧)
5251orcd 869 . . . . . . . 8 (𝑧 = 𝐵 → (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅))
53 trud 1538 . . . . . . . 8 (𝑧 = 𝐵 → ⊤)
5452, 532thd 266 . . . . . . 7 (𝑧 = 𝐵 → ((𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅) ↔ ⊤))
5549, 54ralprg 4624 . . . . . 6 ((𝐴𝑉𝐵𝑉) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅) ↔ ((𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅) ∧ ⊤)))
56553adant3 1124 . . . . 5 ((𝐴𝑉𝐵𝑉𝐴𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅) ↔ ((𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅) ∧ ⊤)))
5737biantru 530 . . . . . 6 ((𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅) ↔ ((𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅) ∧ ⊤))
5836, 57syl6bb 288 . . . . 5 ((𝐴𝑉𝐵𝑉𝐴𝐵) → ((𝐷𝐸) = ∅ ↔ ((𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅) ∧ ⊤)))
5956, 58bitr4d 283 . . . 4 ((𝐴𝑉𝐵𝑉𝐴𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅) ↔ (𝐷𝐸) = ∅))
6040, 59anbi12d 630 . . 3 ((𝐴𝑉𝐵𝑉𝐴𝐵) → ((∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ∧ ∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅)) ↔ ((𝐷𝐸) = ∅ ∧ (𝐷𝐸) = ∅)))
6120, 60bitrd 280 . 2 ((𝐴𝑉𝐵𝑉𝐴𝐵) → (∀𝑦 ∈ {𝐴, 𝐵}∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅) ↔ ((𝐷𝐸) = ∅ ∧ (𝐷𝐸) = ∅)))
62 disjors 5038 . 2 (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ ∀𝑦 ∈ {𝐴, 𝐵}∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅))
63 pm4.24 564 . 2 ((𝐷𝐸) = ∅ ↔ ((𝐷𝐸) = ∅ ∧ (𝐷𝐸) = ∅))
6461, 62, 633bitr4g 315 1 ((𝐴𝑉𝐵𝑉𝐴𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ (𝐷𝐸) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wtru 1529  wcel 2105  wne 3013  wral 3135  csb 3880  cin 3932  c0 4288  {cpr 4559  Disj wdisj 5022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-nul 4289  df-sn 4558  df-pr 4560  df-disj 5023
This theorem is referenced by:  disjdifprg  30253  pmeasmono  31481  probun  31576  meadjun  42621
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