Theorem unissd 3813

Description: Subclass relationship for subclass union. Deduction form of uniss 3810. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
unissd.1	|- ( ph -> A C_ B )

Assertion

Ref	Expression
unissd	|- ( ph -> U. A C_ U. B )

Proof of Theorem unissd

Step	Hyp	Ref	Expression
1		unissd.1	. 2 |- ( ph -> A C_ B )
2		uniss 3810	. 2 |- ( A C_ B -> U. A C_ U. B )
3	1, 2	syl 14	1 |- ( ph -> U. A C_ U. B )

Syntax hints:   -> wi 4   C_ wss 3116   U.cuni 3789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-uni 3790

This theorem is referenced by:  iotanul  5168  tfrlemibfn  6296  tfrlemiubacc  6298  tfr1onlemssrecs  6307  tfr1onlembfn  6312  tfr1onlemubacc  6314  tfrcllemssrecs  6320  tfrcllembfn  6325  tfrcllemubacc  6327  fiuni  6943  eltg3i  12706  unitg  12712  tgss  12713  ntrss  12769