Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssun4 Structured version   Visualization version   GIF version

Theorem ssun4 4105
 Description: Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
Assertion
Ref Expression
ssun4 (𝐴𝐵𝐴 ⊆ (𝐶𝐵))

Proof of Theorem ssun4
StepHypRef Expression
1 ssun2 4103 . 2 𝐵 ⊆ (𝐶𝐵)
2 sstr2 3924 . 2 (𝐴𝐵 → (𝐵 ⊆ (𝐶𝐵) → 𝐴 ⊆ (𝐶𝐵)))
31, 2mpi 20 1 (𝐴𝐵𝐴 ⊆ (𝐶𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∪ cun 3881   ⊆ wss 3883 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3444  df-un 3888  df-in 3890  df-ss 3900 This theorem is referenced by:  ssun  4119  xpsspw  5650  uncmp  22049  volcn  24251  bnj1408  32484  bnj1452  32500  dftrpred3g  33255  pibt2  34985  elrfi  39806  cnvrcl0  40496
 Copyright terms: Public domain W3C validator