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Theorem unss1 4157
 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 3963 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21orim1d 962 . . 3 (𝐴𝐵 → ((𝑥𝐴𝑥𝐶) → (𝑥𝐵𝑥𝐶)))
3 elun 4127 . . 3 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
4 elun 4127 . . 3 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
52, 3, 43imtr4g 298 . 2 (𝐴𝐵 → (𝑥 ∈ (𝐴𝐶) → 𝑥 ∈ (𝐵𝐶)))
65ssrdv 3975 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 843   ∈ wcel 2114   ∪ cun 3936   ⊆ wss 3938 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-v 3498  df-un 3943  df-in 3945  df-ss 3954 This theorem is referenced by:  unss2  4159  unss12  4160  eldifpw  7492  tposss  7895  dftpos4  7913  hashbclem  13813  incexclem  15193  mreexexlem2d  16918  catcoppccl  17370  neitr  21790  restntr  21792  leordtval2  21822  cmpcld  22012  uniioombllem3  24188  limcres  24486  plyss  24791  shlej1  29139  ss2mcls  32817  orderseqlem  33096  noetalem4  33222  bj-rrhatsscchat  34520  pclfinclN  37088
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