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Theorem unss1 4195
Description: Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
unss1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Proof of Theorem unss1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssel 3989 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21orim1d 967 . . 3 (𝐴𝐵 → ((𝑥𝐴𝑥𝐶) → (𝑥𝐵𝑥𝐶)))
3 elun 4163 . . 3 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
4 elun 4163 . . 3 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
52, 3, 43imtr4g 296 . 2 (𝐴𝐵 → (𝑥 ∈ (𝐴𝐶) → 𝑥 ∈ (𝐵𝐶)))
65ssrdv 4001 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 847  wcel 2106  cun 3961  wss 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980
This theorem is referenced by:  unss2  4197  unss12  4198  eldifpw  7787  orderseqlem  8181  tposss  8251  dftpos4  8269  hashbclem  14488  incexclem  15869  mreexexlem2d  17690  catcoppccl  18171  catcoppcclOLD  18172  neitr  23204  restntr  23206  leordtval2  23236  cmpcld  23426  uniioombllem3  25634  limcres  25936  plyss  26253  mulsproplem13  28169  mulsproplem14  28170  shlej1  31389  fineqvac  35090  ss2mcls  35553  bj-rrhatsscchat  37219  pclfinclN  39933
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