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Theorem unss2 4197
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 4195 . 2 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 uncom 4168 . 2 (𝐶𝐴) = (𝐴𝐶)
3 uncom 4168 . 2 (𝐶𝐵) = (𝐵𝐶)
41, 2, 33sstr4g 4041 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cun 3961  wss 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980
This theorem is referenced by:  unss12  4198  ord3ex  5393  xpider  8827  fin1a2lem13  10450  canthp1lem2  10691  seqexw  14055  uniioombllem3  25634  volcn  25655  dvres2lem  25960  mulsproplem13  28169  mulsproplem14  28170  bnj1413  35028  bnj1408  35029
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