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Theorem disjss3 4576
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 2900 . . . . . . 7 (∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅ ↔ ∀𝑥(𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅))
2 simprr 791 . . . . . . . . . . . 12 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → 𝑦𝐶)
3 n0i 3878 . . . . . . . . . . . 12 (𝑦𝐶 → ¬ 𝐶 = ∅)
42, 3syl 17 . . . . . . . . . . 11 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → ¬ 𝐶 = ∅)
5 simpl 471 . . . . . . . . . . . . 13 ((𝑥𝐵𝑦𝐶) → 𝑥𝐵)
65adantl 480 . . . . . . . . . . . 12 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → 𝑥𝐵)
7 eldif 3549 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
8 simpl 471 . . . . . . . . . . . . 13 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → (𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅))
97, 8syl5bir 231 . . . . . . . . . . . 12 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → ((𝑥𝐵 ∧ ¬ 𝑥𝐴) → 𝐶 = ∅))
106, 9mpand 706 . . . . . . . . . . 11 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → (¬ 𝑥𝐴𝐶 = ∅))
114, 10mt3d 138 . . . . . . . . . 10 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → 𝑥𝐴)
1211, 2jca 552 . . . . . . . . 9 (((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) ∧ (𝑥𝐵𝑦𝐶)) → (𝑥𝐴𝑦𝐶))
1312ex 448 . . . . . . . 8 ((𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) → ((𝑥𝐵𝑦𝐶) → (𝑥𝐴𝑦𝐶)))
1413alimi 1729 . . . . . . 7 (∀𝑥(𝑥 ∈ (𝐵𝐴) → 𝐶 = ∅) → ∀𝑥((𝑥𝐵𝑦𝐶) → (𝑥𝐴𝑦𝐶)))
151, 14sylbi 205 . . . . . 6 (∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅ → ∀𝑥((𝑥𝐵𝑦𝐶) → (𝑥𝐴𝑦𝐶)))
16 moim 2506 . . . . . 6 (∀𝑥((𝑥𝐵𝑦𝐶) → (𝑥𝐴𝑦𝐶)) → (∃*𝑥(𝑥𝐴𝑦𝐶) → ∃*𝑥(𝑥𝐵𝑦𝐶)))
1715, 16syl 17 . . . . 5 (∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅ → (∃*𝑥(𝑥𝐴𝑦𝐶) → ∃*𝑥(𝑥𝐵𝑦𝐶)))
1817alimdv 1831 . . . 4 (∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅ → (∀𝑦∃*𝑥(𝑥𝐴𝑦𝐶) → ∀𝑦∃*𝑥(𝑥𝐵𝑦𝐶)))
19 dfdisj2 4549 . . . 4 (Disj 𝑥𝐴 𝐶 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐶))
20 dfdisj2 4549 . . . 4 (Disj 𝑥𝐵 𝐶 ↔ ∀𝑦∃*𝑥(𝑥𝐵𝑦𝐶))
2118, 19, 203imtr4g 283 . . 3 (∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅ → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
2221adantl 480 . 2 ((𝐴𝐵 ∧ ∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅) → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
23 disjss1 4553 . . 3 (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
2423adantr 479 . 2 ((𝐴𝐵 ∧ ∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅) → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
2522, 24impbid 200 1 ((𝐴𝐵 ∧ ∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅) → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wcel 1976  ∃*wmo 2458  wral 2895  cdif 3536  wss 3539  c0 3873  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 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  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-ral 2900  df-rmo 2903  df-v 3174  df-dif 3542  df-in 3546  df-ss 3553  df-nul 3874  df-disj 4548
This theorem is referenced by:  carsggect  29513
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