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Theorem unss2 3160
Description: Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)
Assertion
Ref Expression
unss2 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))

Proof of Theorem unss2
StepHypRef Expression
1 unss1 3158 . 2 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 uncom 3133 . 2 (𝐶𝐴) = (𝐴𝐶)
3 uncom 3133 . 2 (𝐶𝐵) = (𝐵𝐶)
41, 2, 33sstr4g 3056 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  cun 2986  wss 2988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992  df-in 2994  df-ss 3001
This theorem is referenced by:  unss12  3161  difdif2ss  3245  difdifdirss  3354  ord3ex  4001  rdgss  6104  xpiderm  6317
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