Theorem ssuniint 39250

Description: Sufficient condition for being a subclass of the union of an intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypotheses

Ref	Expression
ssuniint.x	|- F/ x ph
ssuniint.a	|- ( ph -> A e. V )
ssuniint.b	|- ( ( ph /\ x e. B ) -> A e. x )

Assertion

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

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem ssuniint

Step	Hyp	Ref	Expression
1		ssuniint.x	. . 3 |- F/ x ph
2		ssuniint.a	. . 3 |- ( ph -> A e. V )
3		ssuniint.b	. . 3 |- ( ( ph /\ x e. B ) -> A e. x )
4	1, 2, 3	elintd 39245	. 2 |- ( ph -> A e. |^| B )
5		elssuni 4467	. 2 |- ( A e. |^| B -> A C_ U. |^| B )
6	4, 5	syl 17	1 |- ( ph -> A C_ U. |^| B )

Syntax hints:   -> wi 4   /\ wa 384   F/wnf 1708   e. wcel 1990   C_ wss 3574   U.cuni 4436   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-in 3581  df-ss 3588  df-uni 4437  df-int 4476

This theorem is referenced by: (None)