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Theorem unss1 3089
 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 2916 . . . 4 (AB → (x Ax B))
21orim1d 688 . . 3 (AB → ((x A x 𝐶) → (x B x 𝐶)))
3 elun 3061 . . 3 (x (A𝐶) ↔ (x A x 𝐶))
4 elun 3061 . . 3 (x (B𝐶) ↔ (x B x 𝐶))
52, 3, 43imtr4g 194 . 2 (AB → (x (A𝐶) → x (B𝐶)))
65ssrdv 2928 1 (AB → (A𝐶) ⊆ (B𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 616   ∈ wcel 1374   ∪ cun 2892   ⊆ wss 2894 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908 This theorem is referenced by:  unss2  3091  unss12  3092  undif1ss  3275  eldifpw  4158  tposss  5783  dftpos4  5800
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