Theorem rabrsndc 3734

Description: A class abstraction over a decidable proposition restricted to a singleton is either the empty set or the singleton itself. (Contributed by Jim Kingdon, 8-Aug-2018.)

Hypotheses

Ref	Expression
rabrsndc.1	|- A e. _V
rabrsndc.2	|- DECID ph

Assertion

Ref	Expression
rabrsndc	|- ( M = { x e. { A } | ph } -> ( M = (/) \/ M = { A } ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   M( x)

Proof of Theorem rabrsndc

Step	Hyp	Ref	Expression
1		rabrsndc.1	. . . . . 6 |- A e. _V
2		rabrsndc.2	. . . . . . . 8 |- DECID ph
3		pm2.1dc 842	. . . . . . . 8 |- (DECID ph -> ( -. ph \/ ph ) )
4	2, 3	ax-mp 5	. . . . . . 7 |- ( -. ph \/ ph )
5	4	sbcth 3042	. . . . . 6 |- ( A e. _V -> [. A / x ]. ( -. ph \/ ph ) )
6	1, 5	ax-mp 5	. . . . 5 |- [. A / x ]. ( -. ph \/ ph )
7		sbcor 3073	. . . . 5 |- ( [. A / x ]. ( -. ph \/ ph ) <-> ( [. A / x ]. -. ph \/ [. A / x ]. ph ) )
8	6, 7	mpbi 145	. . . 4 |- ( [. A / x ]. -. ph \/ [. A / x ]. ph )
9		ralsns 3704	. . . . . 6 |- ( A e. _V -> ( A. x e. { A } -. ph <-> [. A / x ]. -. ph ) )
10	1, 9	ax-mp 5	. . . . 5 |- ( A. x e. { A } -. ph <-> [. A / x ]. -. ph )
11		ralsns 3704	. . . . . 6 |- ( A e. _V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )
12	1, 11	ax-mp 5	. . . . 5 |- ( A. x e. { A } ph <-> [. A / x ]. ph )
13	10, 12	orbi12i 769	. . . 4 |- ( ( A. x e. { A } -. ph \/ A. x e. { A } ph ) <-> ( [. A / x ]. -. ph \/ [. A / x ]. ph ) )
14	8, 13	mpbir 146	. . 3 |- ( A. x e. { A } -. ph \/ A. x e. { A } ph )
15		rabeq0 3521	. . . 4 |- ( { x e. { A } | ph } = (/) <-> A. x e. { A } -. ph )
16		eqcom 2231	. . . . 5 |- ( { x e. { A } | ph } = { A } <-> { A } = { x e. { A } | ph } )
17		rabid2 2708	. . . . 5 |- ( { A } = { x e. { A } | ph } <-> A. x e. { A } ph )
18	16, 17	bitri 184	. . . 4 |- ( { x e. { A } | ph } = { A } <-> A. x e. { A } ph )
19	15, 18	orbi12i 769	. . 3 |- ( ( { x e. { A } | ph } = (/) \/ { x e. { A } | ph } = { A } ) <-> ( A. x e. { A } -. ph \/ A. x e. { A } ph ) )
20	14, 19	mpbir 146	. 2 |- ( { x e. { A } | ph } = (/) \/ { x e. { A } | ph } = { A } )
21		eqeq1 2236	. . 3 |- ( M = { x e. { A } | ph } -> ( M = (/) <-> { x e. { A } | ph } = (/) ) )
22		eqeq1 2236	. . 3 |- ( M = { x e. { A } | ph } -> ( M = { A } <-> { x e. { A } | ph } = { A } ) )
23	21, 22	orbi12d 798	. 2 |- ( M = { x e. { A } | ph } -> ( ( M = (/) \/ M = { A } ) <-> ( { x e. { A } | ph } = (/) \/ { x e. { A } | ph } = { A } ) ) )
24	20, 23	mpbiri 168	1 |- ( M = { x e. { A } | ph } -> ( M = (/) \/ M = { A } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   A.wral 2508   {crab 2512   _Vcvv 2799   [.wsbc 3028   (/)c0 3491   {csn 3666

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-in1 617  ax-in2 618  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-dc 840  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-nul 3492  df-sn 3672

This theorem is referenced by: (None)