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Theorem ssundif 4351
 Description: A condition equivalent to inclusion in the union of two classes. (Contributed by NM, 26-Mar-2007.)
Assertion
Ref Expression
ssundif (𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)

Proof of Theorem ssundif
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pm5.6 996 . . . 4 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥𝐶) ↔ (𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
2 eldif 3873 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
32imbi1i 351 . . . 4 ((𝑥 ∈ (𝐴𝐵) → 𝑥𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥𝐶))
4 elun 4050 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
54imbi2i 337 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
61, 3, 53bitr4ri 305 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
76albii 1801 . 2 (∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
8 dfss2 3881 . 2 (𝐴 ⊆ (𝐵𝐶) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)))
9 dfss2 3881 . 2 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
107, 8, 93bitr4i 304 1 (𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842  ∀wal 1520   ∈ wcel 2081   ∖ cdif 3860   ∪ cun 3861   ⊆ wss 3863 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-v 3439  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878 This theorem is referenced by:  difcom  4352  uneqdifeq  4356  ssunsn2  4671  elpwun  7353  soex  7487  ressuppssdif  7707  frfi  8614  cantnfp1lem3  8994  dfacfin7  9672  zornn0g  9778  fpwwe2lem13  9915  hashbclem  13663  incexclem  15029  ramub1lem1  16196  lpcls  21661  cmpcld  21699  alexsubALTlem3  22346  restmetu  22868  uniiccdif  23867  abelthlem2  24708  abelthlem3  24709  pmtrcnelor  30399  imadifss  34423  frege124d  39616
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