Theorem unrab 3208

Description: Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)

Assertion

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

Proof of Theorem unrab

Step	Hyp	Ref	Expression
1		df-rab 2315	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		df-rab 2315	. . 3 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
3	1, 2	uneq12i 3095	. 2 |- ( { x e. A | ph } u. { x e. A | ps } ) = ( { x | ( x e. A /\ ph ) } u. { x | ( x e. A /\ ps ) } )
4		df-rab 2315	. . 3 |- { x e. A | ( ph \/ ps ) } = { x | ( x e. A /\ ( ph \/ ps ) ) }
5		unab 3204	. . . 4 |- ( { x | ( x e. A /\ ph ) } u. { x | ( x e. A /\ ps ) } ) = { x | ( ( x e. A /\ ph ) \/ ( x e. A /\ ps ) ) }
6		andi 731	. . . . 5 |- ( ( x e. A /\ ( ph \/ ps ) ) <-> ( ( x e. A /\ ph ) \/ ( x e. A /\ ps ) ) )
7	6	abbii 2153	. . . 4 |- { x | ( x e. A /\ ( ph \/ ps ) ) } = { x | ( ( x e. A /\ ph ) \/ ( x e. A /\ ps ) ) }
8	5, 7	eqtr4i 2063	. . 3 |- ( { x | ( x e. A /\ ph ) } u. { x | ( x e. A /\ ps ) } ) = { x | ( x e. A /\ ( ph \/ ps ) ) }
9	4, 8	eqtr4i 2063	. 2 |- { x e. A | ( ph \/ ps ) } = ( { x | ( x e. A /\ ph ) } u. { x | ( x e. A /\ ps ) } )
10	3, 9	eqtr4i 2063	1 |- ( { x e. A | ph } u. { x e. A | ps } ) = { x e. A | ( ph \/ ps ) }

Syntax hints:   /\ wa 97   \/ wo 629   = wceq 1243   e. wcel 1393   {cab 2026   {crab 2310   u. cun 2915

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-un 2922

This theorem is referenced by:  rabxmdc  3249