Theorem bj-unrab 32922

Description: Generalization of unrab 3898. Equality need not hold. (Contributed by BJ, 21-Apr-2019.)

Assertion

Ref	Expression
bj-unrab	|- ( { x e. A | ph } u. { x e. B | ps } ) C_ { x e. ( A u. B ) | ( ph \/ ps ) }

Distinct variable groups:   x, A   x, B

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

Proof of Theorem bj-unrab

Step	Hyp	Ref	Expression
1		ssun1 3776	. . . 4 |- A C_ ( A u. B )
2		rabss2 3685	. . . 4 |- ( A C_ ( A u. B ) -> { x e. A | ph } C_ { x e. ( A u. B ) | ph } )
3	1, 2	ax-mp 5	. . 3 |- { x e. A | ph } C_ { x e. ( A u. B ) | ph }
4		orc 400	. . . . 5 |- ( ph -> ( ph \/ ps ) )
5	4	a1i 11	. . . 4 |- ( x e. ( A u. B ) -> ( ph -> ( ph \/ ps ) ) )
6	5	ss2rabi 3684	. . 3 |- { x e. ( A u. B ) | ph } C_ { x e. ( A u. B ) | ( ph \/ ps ) }
7	3, 6	sstri 3612	. 2 |- { x e. A | ph } C_ { x e. ( A u. B ) | ( ph \/ ps ) }
8		ssun2 3777	. . . 4 |- B C_ ( A u. B )
9		rabss2 3685	. . . 4 |- ( B C_ ( A u. B ) -> { x e. B | ps } C_ { x e. ( A u. B ) | ps } )
10	8, 9	ax-mp 5	. . 3 |- { x e. B | ps } C_ { x e. ( A u. B ) | ps }
11		olc 399	. . . . 5 |- ( ps -> ( ph \/ ps ) )
12	11	a1i 11	. . . 4 |- ( x e. ( A u. B ) -> ( ps -> ( ph \/ ps ) ) )
13	12	ss2rabi 3684	. . 3 |- { x e. ( A u. B ) | ps } C_ { x e. ( A u. B ) | ( ph \/ ps ) }
14	10, 13	sstri 3612	. 2 |- { x e. B | ps } C_ { x e. ( A u. B ) | ( ph \/ ps ) }
15	7, 14	unssi 3788	1 |- ( { x e. A | ph } u. { x e. B | ps } ) C_ { x e. ( A u. B ) | ( ph \/ ps ) }

Syntax hints:   -> wi 4   \/ wo 383   e. wcel 1990   {crab 2916   u. cun 3572   C_ wss 3574

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-rab 2921  df-v 3202  df-un 3579  df-in 3581  df-ss 3588

This theorem is referenced by: (None)