Theorem unrab 3475

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 2517	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		df-rab 2517	. . 3 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
3	1, 2	uneq12i 3356	. 2 |- ( { x e. A | ph } u. { x e. A | ps } ) = ( { x | ( x e. A /\ ph ) } u. { x | ( x e. A /\ ps ) } )
4		df-rab 2517	. . 3 |- { x e. A | ( ph \/ ps ) } = { x | ( x e. A /\ ( ph \/ ps ) ) }
5		unab 3471	. . . 4 |- ( { x | ( x e. A /\ ph ) } u. { x | ( x e. A /\ ps ) } ) = { x | ( ( x e. A /\ ph ) \/ ( x e. A /\ ps ) ) }
6		andi 823	. . . . 5 |- ( ( x e. A /\ ( ph \/ ps ) ) <-> ( ( x e. A /\ ph ) \/ ( x e. A /\ ps ) ) )
7	6	abbii 2345	. . . 4 |- { x | ( x e. A /\ ( ph \/ ps ) ) } = { x | ( ( x e. A /\ ph ) \/ ( x e. A /\ ps ) ) }
8	5, 7	eqtr4i 2253	. . 3 |- ( { x | ( x e. A /\ ph ) } u. { x | ( x e. A /\ ps ) } ) = { x | ( x e. A /\ ( ph \/ ps ) ) }
9	4, 8	eqtr4i 2253	. 2 |- { x e. A | ( ph \/ ps ) } = ( { x | ( x e. A /\ ph ) } u. { x | ( x e. A /\ ps ) } )
10	3, 9	eqtr4i 2253	1 |- ( { x e. A | ph } u. { x e. A | ps } ) = { x e. A | ( ph \/ ps ) }

Syntax hints:   /\ wa 104   \/ wo 713   = wceq 1395   e. wcel 2200   {cab 2215   {crab 2512   u. cun 3195

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-un 3201

This theorem is referenced by:  rabxmdc  3523  phiprmpw  12739  unennn  12963  znnen  12964  lgsquadlem2  15751