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Theorem unss2 4187
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 4185 . 2 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 uncom 4158 . 2 (𝐶𝐴) = (𝐴𝐶)
3 uncom 4158 . 2 (𝐶𝐵) = (𝐵𝐶)
41, 2, 33sstr4g 4037 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cun 3949  wss 3951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968
This theorem is referenced by:  unss12  4188  ord3ex  5387  xpider  8828  fin1a2lem13  10452  canthp1lem2  10693  seqexw  14058  uniioombllem3  25620  volcn  25641  dvres2lem  25945  mulsproplem13  28154  mulsproplem14  28155  bnj1413  35049  bnj1408  35050
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