Theorem unissd 3959

Description: Subclass relationship for subclass union. Deduction form of uniss 3956. (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 3956	. 2 |- ( A C_ B -> U. A C_ U. B )
3	1, 2	syl 14	1 |- ( ph -> U. A C_ U. B )

This proof depends on syntax axioms:   -> wi 4   C_ wss 3220   U.cuni 3935

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-uni 3936

This theorem is used by:  iotanul  5353  tfrlemibfn  6599  tfrlemiubacc  6601  tfr1onlemssrecs  6610  tfr1onlembfn  6615  tfr1onlemubacc  6617  tfrcllemssrecs  6623  tfrcllembfn  6628  tfrcllemubacc  6630  fiuni  7312  eltg3i  15157  unitg  15163  tgss  15164  ntrss  15220