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Theorem disjss3 4684
Description: Expand a disjoint collection with any number of empty sets. (Contributed by Mario Carneiro, 15-Nov-2016.)
Assertion
Ref Expression
disjss3 ((𝐴𝐵 ∧ ∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅) → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem disjss3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-ral 2946 . . . . . . 7 (∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅ ↔ ∀𝑥(𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅))
2 simprr 811 . . . . . . . . . . . 12 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → 𝑦𝐶)
3 n0i 3953 . . . . . . . . . . . 12 (𝑦𝐶 → ¬ 𝐶 = ∅)
42, 3syl 17 . . . . . . . . . . 11 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → ¬ 𝐶 = ∅)
5 simpl 472 . . . . . . . . . . . . 13 ((𝑥𝐵𝑦𝐶) → 𝑥𝐵)
65adantl 481 . . . . . . . . . . . 12 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → 𝑥𝐵)
7 eldif 3617 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
8 simpl 472 . . . . . . . . . . . . 13 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → (𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅))
97, 8syl5bir 233 . . . . . . . . . . . 12 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → ((𝑥𝐵 ∧ ¬ 𝑥𝐴) → 𝐶 = ∅))
106, 9mpand 711 . . . . . . . . . . 11 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → (¬ 𝑥𝐴𝐶 = ∅))
114, 10mt3d 140 . . . . . . . . . 10 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → 𝑥𝐴)
1211, 2jca 553 . . . . . . . . 9 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → (𝑥𝐴𝑦𝐶))
1312ex 449 . . . . . . . 8 ((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) → ((𝑥𝐵𝑦𝐶) → (𝑥𝐴𝑦𝐶)))
1413alimi 1779 . . . . . . 7 (∀𝑥(𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) → ∀𝑥((𝑥𝐵𝑦𝐶) → (𝑥𝐴𝑦𝐶)))
151, 14sylbi 207 . . . . . 6 (∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅ → ∀𝑥((𝑥𝐵𝑦𝐶) → (𝑥𝐴𝑦𝐶)))
16 moim 2548 . . . . . 6 (∀𝑥((𝑥𝐵𝑦𝐶) → (𝑥𝐴𝑦𝐶)) → (∃*𝑥(𝑥𝐴𝑦𝐶) → ∃*𝑥(𝑥𝐵𝑦𝐶)))
1715, 16syl 17 . . . . 5 (∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅ → (∃*𝑥(𝑥𝐴𝑦𝐶) → ∃*𝑥(𝑥𝐵𝑦𝐶)))
1817alimdv 1885 . . . 4 (∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅ → (∀𝑦∃*𝑥(𝑥𝐴𝑦𝐶) → ∀𝑦∃*𝑥(𝑥𝐵𝑦𝐶)))
19 dfdisj2 4654 . . . 4 (Disj 𝑥𝐴 𝐶 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐶))
20 dfdisj2 4654 . . . 4 (Disj 𝑥𝐵 𝐶 ↔ ∀𝑦∃*𝑥(𝑥𝐵𝑦𝐶))
2118, 19, 203imtr4g 285 . . 3 (∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅ → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
2221adantl 481 . 2 ((𝐴𝐵 ∧ ∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅) → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
23 disjss1 4658 . . 3 (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
2423adantr 480 . 2 ((𝐴𝐵 ∧ ∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅) → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
2522, 24impbid 202 1 ((𝐴𝐵 ∧ ∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅) → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wcel 2030  ∃*wmo 2499  wral 2941  cdif 3604  wss 3607  c0 3948  Disj wdisj 4652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rmo 2949  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-disj 4653
This theorem is referenced by:  carsggect  30508
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