Theorem rabxm 3961

Description: Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)

Assertion

Ref	Expression
rabxm	|- A = ( { x e. A | ph } u. { x e. A | -. ph } )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem rabxm

Step	Hyp	Ref	Expression
1		rabid2 3118	. . 3 |- ( A = { x e. A | ( ph \/ -. ph ) } <-> A. x e. A ( ph \/ -. ph ) )
2		exmidd 432	. . 3 |- ( x e. A -> ( ph \/ -. ph ) )
3	1, 2	mprgbir 2927	. 2 |- A = { x e. A | ( ph \/ -. ph ) }
4		unrab 3898	. 2 |- ( { x e. A | ph } u. { x e. A | -. ph } ) = { x e. A | ( ph \/ -. ph ) }
5	3, 4	eqtr4i 2647	1 |- A = ( { x e. A | ph } u. { x e. A | -. ph } )

Syntax hints:   -. wn 3   \/ wo 383   = wceq 1483   e. wcel 1990   {crab 2916   u. cun 3572

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

This theorem is referenced by:  elnelun  3964  elnelunOLD  3966  vtxdgoddnumeven  26449  esumrnmpt2  30130  ddemeas  30299  ballotth  30599  mbfposadd  33457  jm2.22  37562