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Theorem ssundif 4488
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 1001 . . . 4 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥𝐶) ↔ (𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
2 eldif 3959 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
32imbi1i 350 . . . 4 ((𝑥 ∈ (𝐴𝐵) → 𝑥𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥𝐶))
4 elun 4149 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
54imbi2i 336 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
61, 3, 53bitr4ri 304 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
76albii 1822 . 2 (∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
8 dfss2 3969 . 2 (𝐴 ⊆ (𝐵𝐶) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)))
9 dfss2 3969 . 2 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
107, 8, 93bitr4i 303 1 (𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846  wal 1540  wcel 2107  cdif 3946  cun 3947  wss 3949
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966
This theorem is referenced by:  difcom  4489  uneqdifeq  4493  ssunsn2  4831  elpwun  7756  soex  7912  ressuppssdif  8170  ssfi  9173  frfi  9288  cantnfp1lem3  9675  dfacfin7  10394  zornn0g  10500  fpwwe2lem12  10637  hashbclem  14411  incexclem  15782  ramub1lem1  16959  lpcls  22868  cmpcld  22906  alexsubALTlem3  23553  restmetu  24079  uniiccdif  25095  abelthlem2  25944  abelthlem3  25945  pmtrcnelor  32252  imadifss  36463  frege124d  42512
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