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Theorem unss2 3114
 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 3112 . 2 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 uncom 3087 . 2 (𝐶𝐴) = (𝐴𝐶)
3 uncom 3087 . 2 (𝐶𝐵) = (𝐵𝐶)
41, 2, 33sstr4g 2986 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∪ cun 2915   ⊆ wss 2917 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931 This theorem is referenced by:  unss12  3115  difdif2ss  3194  difdifdirss  3307  ord3ex  3941  rdgss  5970  xpiderm  6177
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