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Theorem unss2 3928
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 3926 . 2 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 uncom 3901 . 2 (𝐶𝐴) = (𝐴𝐶)
3 uncom 3901 . 2 (𝐶𝐵) = (𝐵𝐶)
41, 2, 33sstr4g 3788 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cun 3714  wss 3716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-v 3343  df-un 3721  df-in 3723  df-ss 3730
This theorem is referenced by:  unss12  3929  ord3ex  5006  xpider  7988  fin1a2lem13  9447  canthp1lem2  9688  uniioombllem3  23574  volcn  23595  dvres2lem  23894  bnj1413  31432  bnj1408  31433
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