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Theorem disjprg 5032
 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 2762 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 = 𝑧𝐴 = 𝑧))
2 nfcv 2919 . . . . . . . . . 10 𝑥𝐴
3 nfcv 2919 . . . . . . . . . 10 𝑥𝐷
4 disjprg.1 . . . . . . . . . 10 (𝑥 = 𝐴𝐶 = 𝐷)
52, 3, 4csbhypf 3835 . . . . . . . . 9 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐷)
65ineq1d 4118 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = (𝐷𝑧 / 𝑥𝐶))
76eqeq1d 2760 . . . . . . 7 (𝑦 = 𝐴 → ((𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅ ↔ (𝐷𝑧 / 𝑥𝐶) = ∅))
81, 7orbi12d 916 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅) ↔ (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅)))
98ralbidv 3126 . . . . 5 (𝑦 = 𝐴 → (∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅) ↔ ∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅)))
10 eqeq1 2762 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 = 𝑧𝐵 = 𝑧))
11 nfcv 2919 . . . . . . . . . 10 𝑥𝐵
12 nfcv 2919 . . . . . . . . . 10 𝑥𝐸
13 disjprg.2 . . . . . . . . . 10 (𝑥 = 𝐵𝐶 = 𝐸)
1411, 12, 13csbhypf 3835 . . . . . . . . 9 (𝑦 = 𝐵𝑦 / 𝑥𝐶 = 𝐸)
1514ineq1d 4118 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = (𝐸𝑧 / 𝑥𝐶))
1615eqeq1d 2760 . . . . . . 7 (𝑦 = 𝐵 → ((𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅ ↔ (𝐸𝑧 / 𝑥𝐶) = ∅))
1710, 16orbi12d 916 . . . . . 6 (𝑦 = 𝐵 → ((𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅) ↔ (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅)))
1817ralbidv 3126 . . . . 5 (𝑦 = 𝐵 → (∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅) ↔ ∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅)))
199, 18ralprg 4592 . . . 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 1129 . . 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 2764 . . . . . . . . 9 (𝑧 = 𝐴𝐴 = 𝑧)
2322orcd 870 . . . . . . . 8 (𝑧 = 𝐴 → (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅))
24 trud 1548 . . . . . . . 8 (𝑧 = 𝐴 → ⊤)
2523, 242thd 268 . . . . . . 7 (𝑧 = 𝐴 → ((𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ↔ ⊤))
26 eqeq2 2770 . . . . . . . 8 (𝑧 = 𝐵 → (𝐴 = 𝑧𝐴 = 𝐵))
2711, 12, 13csbhypf 3835 . . . . . . . . . 10 (𝑧 = 𝐵𝑧 / 𝑥𝐶 = 𝐸)
2827ineq2d 4119 . . . . . . . . 9 (𝑧 = 𝐵 → (𝐷𝑧 / 𝑥𝐶) = (𝐷𝐸))
2928eqeq1d 2760 . . . . . . . 8 (𝑧 = 𝐵 → ((𝐷𝑧 / 𝑥𝐶) = ∅ ↔ (𝐷𝐸) = ∅))
3026, 29orbi12d 916 . . . . . . 7 (𝑧 = 𝐵 → ((𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ↔ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅)))
3125, 30ralprg 4592 . . . . . 6 ((𝐴𝑉𝐵𝑉) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅))))
32313adant3 1129 . . . . 5 ((𝐴𝑉𝐵𝑉𝐴𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅))))
33 simp3 1135 . . . . . . . 8 ((𝐴𝑉𝐵𝑉𝐴𝐵) → 𝐴𝐵)
3433neneqd 2956 . . . . . . 7 ((𝐴𝑉𝐵𝑉𝐴𝐵) → ¬ 𝐴 = 𝐵)
35 biorf 934 . . . . . . 7 𝐴 = 𝐵 → ((𝐷𝐸) = ∅ ↔ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅)))
3634, 35syl 17 . . . . . 6 ((𝐴𝑉𝐵𝑉𝐴𝐵) → ((𝐷𝐸) = ∅ ↔ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅)))
37 tru 1542 . . . . . . 7
3837biantrur 534 . . . . . 6 ((𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅) ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅)))
3936, 38bitrdi 290 . . . . 5 ((𝐴𝑉𝐵𝑉𝐴𝐵) → ((𝐷𝐸) = ∅ ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅))))
4032, 39bitr4d 285 . . . 4 ((𝐴𝑉𝐵𝑉𝐴𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ↔ (𝐷𝐸) = ∅))
41 eqeq2 2770 . . . . . . . . 9 (𝑧 = 𝐴 → (𝐵 = 𝑧𝐵 = 𝐴))
42 eqcom 2765 . . . . . . . . 9 (𝐵 = 𝐴𝐴 = 𝐵)
4341, 42bitrdi 290 . . . . . . . 8 (𝑧 = 𝐴 → (𝐵 = 𝑧𝐴 = 𝐵))
442, 3, 4csbhypf 3835 . . . . . . . . . . 11 (𝑧 = 𝐴𝑧 / 𝑥𝐶 = 𝐷)
4544ineq2d 4119 . . . . . . . . . 10 (𝑧 = 𝐴 → (𝐸𝑧 / 𝑥𝐶) = (𝐸𝐷))
46 incom 4108 . . . . . . . . . 10 (𝐸𝐷) = (𝐷𝐸)
4745, 46eqtrdi 2809 . . . . . . . . 9 (𝑧 = 𝐴 → (𝐸𝑧 / 𝑥𝐶) = (𝐷𝐸))
4847eqeq1d 2760 . . . . . . . 8 (𝑧 = 𝐴 → ((𝐸𝑧 / 𝑥𝐶) = ∅ ↔ (𝐷𝐸) = ∅))
4943, 48orbi12d 916 . . . . . . 7 (𝑧 = 𝐴 → ((𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅) ↔ (𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅)))
50 id 22 . . . . . . . . . 10 (𝑧 = 𝐵𝑧 = 𝐵)
5150eqcomd 2764 . . . . . . . . 9 (𝑧 = 𝐵𝐵 = 𝑧)
5251orcd 870 . . . . . . . 8 (𝑧 = 𝐵 → (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅))
53 trud 1548 . . . . . . . 8 (𝑧 = 𝐵 → ⊤)
5452, 532thd 268 . . . . . . 7 (𝑧 = 𝐵 → ((𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅) ↔ ⊤))
5549, 54ralprg 4592 . . . . . 6 ((𝐴𝑉𝐵𝑉) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅) ↔ ((𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅) ∧ ⊤)))
56553adant3 1129 . . . . 5 ((𝐴𝑉𝐵𝑉𝐴𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅) ↔ ((𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅) ∧ ⊤)))
5737biantru 533 . . . . . 6 ((𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅) ↔ ((𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅) ∧ ⊤))
5836, 57bitrdi 290 . . . . 5 ((𝐴𝑉𝐵𝑉𝐴𝐵) → ((𝐷𝐸) = ∅ ↔ ((𝐴 = 𝐵 ∨ (𝐷𝐸) = ∅) ∧ ⊤)))
5956, 58bitr4d 285 . . . 4 ((𝐴𝑉𝐵𝑉𝐴𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅) ↔ (𝐷𝐸) = ∅))
6040, 59anbi12d 633 . . 3 ((𝐴𝑉𝐵𝑉𝐴𝐵) → ((∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷𝑧 / 𝑥𝐶) = ∅) ∧ ∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸𝑧 / 𝑥𝐶) = ∅)) ↔ ((𝐷𝐸) = ∅ ∧ (𝐷𝐸) = ∅)))
6120, 60bitrd 282 . 2 ((𝐴𝑉𝐵𝑉𝐴𝐵) → (∀𝑦 ∈ {𝐴, 𝐵}∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅) ↔ ((𝐷𝐸) = ∅ ∧ (𝐷𝐸) = ∅)))
62 disjors 5017 . 2 (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ ∀𝑦 ∈ {𝐴, 𝐵}∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐶𝑧 / 𝑥𝐶) = ∅))
63 pm4.24 567 . 2 ((𝐷𝐸) = ∅ ↔ ((𝐷𝐸) = ∅ ∧ (𝐷𝐸) = ∅))
6461, 62, 633bitr4g 317 1 ((𝐴𝑉𝐵𝑉𝐴𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ (𝐷𝐸) = ∅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538  ⊤wtru 1539   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ⦋csb 3807   ∩ cin 3859  ∅c0 4227  {cpr 4527  Disj wdisj 5001 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-nul 4228  df-sn 4526  df-pr 4528  df-disj 5002 This theorem is referenced by:  disjdifprg  30449  pmeasmono  31822  probun  31917  meadjun  43502
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