Theorem unissd 3833

Description: Subclass relationship for subclass union. Deduction form of uniss 3830. (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 3830	. 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 3129   U.cuni 3809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142  df-uni 3810

This theorem is referenced by:  iotanul  5192  tfrlemibfn  6326  tfrlemiubacc  6328  tfr1onlemssrecs  6337  tfr1onlembfn  6342  tfr1onlemubacc  6344  tfrcllemssrecs  6350  tfrcllembfn  6355  tfrcllemubacc  6357  fiuni  6974  eltg3i  13427  unitg  13433  tgss  13434  ntrss  13490