Theorem unrab 3430

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 2481	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		df-rab 2481	. . 3 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
3	1, 2	uneq12i 3311	. 2 |- ( { x e. A | ph } u. { x e. A | ps } ) = ( { x | ( x e. A /\ ph ) } u. { x | ( x e. A /\ ps ) } )
4		df-rab 2481	. . 3 |- { x e. A | ( ph \/ ps ) } = { x | ( x e. A /\ ( ph \/ ps ) ) }
5		unab 3426	. . . 4 |- ( { x | ( x e. A /\ ph ) } u. { x | ( x e. A /\ ps ) } ) = { x | ( ( x e. A /\ ph ) \/ ( x e. A /\ ps ) ) }
6		andi 819	. . . . 5 |- ( ( x e. A /\ ( ph \/ ps ) ) <-> ( ( x e. A /\ ph ) \/ ( x e. A /\ ps ) ) )
7	6	abbii 2309	. . . 4 |- { x | ( x e. A /\ ( ph \/ ps ) ) } = { x | ( ( x e. A /\ ph ) \/ ( x e. A /\ ps ) ) }
8	5, 7	eqtr4i 2217	. . 3 |- ( { x | ( x e. A /\ ph ) } u. { x | ( x e. A /\ ps ) } ) = { x | ( x e. A /\ ( ph \/ ps ) ) }
9	4, 8	eqtr4i 2217	. 2 |- { x e. A | ( ph \/ ps ) } = ( { x | ( x e. A /\ ph ) } u. { x | ( x e. A /\ ps ) } )
10	3, 9	eqtr4i 2217	1 |- ( { x e. A | ph } u. { x e. A | ps } ) = { x e. A | ( ph \/ ps ) }

Syntax hints:   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2164   {cab 2179   {crab 2476   u. cun 3151

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-un 3157

This theorem is referenced by:  rabxmdc  3478  phiprmpw  12360  unennn  12554  znnen  12555