Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  disjss3 Structured version   Visualization version   GIF version

Theorem disjss3 5032
 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 3076 . . . . . . 7 (∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅ ↔ ∀𝑥(𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅))
2 simprr 773 . . . . . . . . . . . 12 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → 𝑦𝐶)
3 n0i 4233 . . . . . . . . . . . 12 (𝑦𝐶 → ¬ 𝐶 = ∅)
42, 3syl 17 . . . . . . . . . . 11 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → ¬ 𝐶 = ∅)
5 simpl 487 . . . . . . . . . . . . 13 ((𝑥𝐵𝑦𝐶) → 𝑥𝐵)
65adantl 486 . . . . . . . . . . . 12 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → 𝑥𝐵)
7 eldif 3869 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
8 simpl 487 . . . . . . . . . . . . 13 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → (𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅))
97, 8syl5bir 246 . . . . . . . . . . . 12 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → ((𝑥𝐵 ∧ ¬ 𝑥𝐴) → 𝐶 = ∅))
106, 9mpand 695 . . . . . . . . . . 11 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → (¬ 𝑥𝐴𝐶 = ∅))
114, 10mt3d 150 . . . . . . . . . 10 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → 𝑥𝐴)
1211, 2jca 516 . . . . . . . . 9 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → (𝑥𝐴𝑦𝐶))
1312ex 417 . . . . . . . 8 ((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) → ((𝑥𝐵𝑦𝐶) → (𝑥𝐴𝑦𝐶)))
1413alimi 1814 . . . . . . 7 (∀𝑥(𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) → ∀𝑥((𝑥𝐵𝑦𝐶) → (𝑥𝐴𝑦𝐶)))
151, 14sylbi 220 . . . . . 6 (∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅ → ∀𝑥((𝑥𝐵𝑦𝐶) → (𝑥𝐴𝑦𝐶)))
16 moim 2562 . . . . . 6 (∀𝑥((𝑥𝐵𝑦𝐶) → (𝑥𝐴𝑦𝐶)) → (∃*𝑥(𝑥𝐴𝑦𝐶) → ∃*𝑥(𝑥𝐵𝑦𝐶)))
1715, 16syl 17 . . . . 5 (∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅ → (∃*𝑥(𝑥𝐴𝑦𝐶) → ∃*𝑥(𝑥𝐵𝑦𝐶)))
1817alimdv 1918 . . . 4 (∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅ → (∀𝑦∃*𝑥(𝑥𝐴𝑦𝐶) → ∀𝑦∃*𝑥(𝑥𝐵𝑦𝐶)))
19 dfdisj2 5000 . . . 4 (Disj 𝑥𝐴 𝐶 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐶))
20 dfdisj2 5000 . . . 4 (Disj 𝑥𝐵 𝐶 ↔ ∀𝑦∃*𝑥(𝑥𝐵𝑦𝐶))
2118, 19, 203imtr4g 300 . . 3 (∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅ → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
2221adantl 486 . 2 ((𝐴𝐵 ∧ ∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅) → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
23 disjss1 5004 . . 3 (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
2423adantr 485 . 2 ((𝐴𝐵 ∧ ∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅) → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
2522, 24impbid 215 1 ((𝐴𝐵 ∧ ∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅) → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1537   = wceq 1539   ∈ wcel 2112  ∃*wmo 2556  ∀wral 3071   ∖ cdif 3856   ⊆ wss 3859  ∅c0 4226  Disj wdisj 4998 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-ex 1783  df-sb 2071  df-mo 2558  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-rmo 3079  df-v 3412  df-dif 3862  df-in 3866  df-ss 3876  df-nul 4227  df-disj 4999 This theorem is referenced by:  carsggect  31805
 Copyright terms: Public domain W3C validator