Theorem unissd 3943

Description: Subclass relationship for subclass union. Deduction form of uniss 3940. (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 3940	. 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 3214   U.cuni 3919

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220  df-ss 3227  df-uni 3920

This theorem is referenced by:  iotanul  5333  tfrlemibfn  6572  tfrlemiubacc  6574  tfr1onlemssrecs  6583  tfr1onlembfn  6588  tfr1onlemubacc  6590  tfrcllemssrecs  6596  tfrcllembfn  6601  tfrcllemubacc  6603  fiuni  7278  eltg3i  15047  unitg  15053  tgss  15054  ntrss  15110