Theorem undisjrab 38505

Description: Union of two disjoint restricted class abstractions; compare unrab 3898. (Contributed by Steve Rodriguez, 28-Feb-2020.)

Assertion

Ref	Expression
undisjrab	|- ( ( { x e. A | ph } i^i { x e. A | ps } ) = (/) <-> ( { x e. A | ph } u. { x e. A | ps } ) = { x e. A | ( ph \/_ ps ) } )

Proof of Theorem undisjrab

Step	Hyp	Ref	Expression
1		rabeq0 3957	. . 3 |- ( { x e. A | ( ph /\ ps ) } = (/) <-> A. x e. A -. ( ph /\ ps ) )
2		df-nan 1448	. . . . 5 |- ( ( ph -/\ ps ) <-> -. ( ph /\ ps ) )
3		nanorxor 38504	. . . . 5 |- ( ( ph -/\ ps ) <-> ( ( ph \/ ps ) <-> ( ph \/_ ps ) ) )
4	2, 3	bitr3i 266	. . . 4 |- ( -. ( ph /\ ps ) <-> ( ( ph \/ ps ) <-> ( ph \/_ ps ) ) )
5	4	ralbii 2980	. . 3 |- ( A. x e. A -. ( ph /\ ps ) <-> A. x e. A ( ( ph \/ ps ) <-> ( ph \/_ ps ) ) )
6		rabbi 3120	. . 3 |- ( A. x e. A ( ( ph \/ ps ) <-> ( ph \/_ ps ) ) <-> { x e. A | ( ph \/ ps ) } = { x e. A | ( ph \/_ ps ) } )
7	1, 5, 6	3bitri 286	. 2 |- ( { x e. A | ( ph /\ ps ) } = (/) <-> { x e. A | ( ph \/ ps ) } = { x e. A | ( ph \/_ ps ) } )
8		inrab 3899	. . 3 |- ( { x e. A | ph } i^i { x e. A | ps } ) = { x e. A | ( ph /\ ps ) }
9	8	eqeq1i 2627	. 2 |- ( ( { x e. A | ph } i^i { x e. A | ps } ) = (/) <-> { x e. A | ( ph /\ ps ) } = (/) )
10		unrab 3898	. . 3 |- ( { x e. A | ph } u. { x e. A | ps } ) = { x e. A | ( ph \/ ps ) }
11	10	eqeq1i 2627	. 2 |- ( ( { x e. A | ph } u. { x e. A | ps } ) = { x e. A | ( ph \/_ ps ) } <-> { x e. A | ( ph \/ ps ) } = { x e. A | ( ph \/_ ps ) } )
12	7, 9, 11	3bitr4i 292	1 |- ( ( { x e. A | ph } i^i { x e. A | ps } ) = (/) <-> ( { x e. A | ph } u. { x e. A | ps } ) = { x e. A | ( ph \/_ ps ) } )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   -/\ wnan 1447   \/_ wxo 1464   = wceq 1483   A.wral 2912   {crab 2916   u. cun 3572   i^i cin 3573   (/)c0 3915

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-nan 1448  df-xor 1465  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-dif 3577  df-un 3579  df-in 3581  df-nul 3916

This theorem is referenced by: (None)