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Theorem ssundif 4429
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 pm5.6 995 . . . 4 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥𝐶) ↔ (𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
2 eldif 3943 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
32imbi1i 351 . . . 4 ((𝑥 ∈ (𝐴𝐵) → 𝑥𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥𝐶))
4 elun 4122 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
54imbi2i 337 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
61, 3, 53bitr4ri 305 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
76albii 1811 . 2 (∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
8 dfss2 3952 . 2 (𝐴 ⊆ (𝐵𝐶) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)))
9 dfss2 3952 . 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 841  wal 1526  wcel 2105  cdif 3930  cun 3931  wss 3933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949
This theorem is referenced by:  difcom  4430  uneqdifeq  4434  ssunsn2  4752  elpwun  7480  soex  7615  ressuppssdif  7840  frfi  8751  cantnfp1lem3  9131  dfacfin7  9809  zornn0g  9915  fpwwe2lem13  10052  hashbclem  13798  incexclem  15179  ramub1lem1  16350  lpcls  21900  cmpcld  21938  alexsubALTlem3  22585  restmetu  23107  uniiccdif  24106  abelthlem2  24947  abelthlem3  24948  pmtrcnelor  30662  imadifss  34748  frege124d  39984
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