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Theorem unss2 4072
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 4070 . 2 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 uncom 4044 . 2 (𝐶𝐴) = (𝐴𝐶)
3 uncom 4044 . 2 (𝐶𝐵) = (𝐵𝐶)
41, 2, 33sstr4g 3923 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cun 3842  wss 3844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-v 3401  df-un 3849  df-in 3851  df-ss 3861
This theorem is referenced by:  unss12  4073  ord3ex  5255  xpider  8402  fin1a2lem13  9915  canthp1lem2  10156  seqexw  13479  uniioombllem3  24340  volcn  24361  dvres2lem  24665  bnj1413  32589  bnj1408  32590
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