Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  unss2 Structured version   GIF version

Theorem unss2 3091
 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 3089 . 2 (AB → (A𝐶) ⊆ (B𝐶))
2 uncom 3064 . 2 (𝐶A) = (A𝐶)
3 uncom 3064 . 2 (𝐶B) = (B𝐶)
41, 2, 33sstr4g 2963 1 (AB → (𝐶A) ⊆ (𝐶B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∪ 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:  unss12  3092  difdif2ss  3171  difdifdirss  3284  ord3ex  3915  rdgss  5890  xpiderm  6088
 Copyright terms: Public domain W3C validator