Theorem rabrsndc 3460

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 778	. . . . . . . 8 |- (DECID ph -> ( -. ph \/ ph ) )
4	2, 3	ax-mp 7	. . . . . . 7 |- ( -. ph \/ ph )
5	4	sbcth 2828	. . . . . 6 |- ( A e. _V -> [. A / x ]. ( -. ph \/ ph ) )
6	1, 5	ax-mp 7	. . . . 5 |- [. A / x ]. ( -. ph \/ ph )
7		sbcor 2858	. . . . 5 |- ( [. A / x ]. ( -. ph \/ ph ) <-> ( [. A / x ]. -. ph \/ [. A / x ]. ph ) )
8	6, 7	mpbi 143	. . . 4 |- ( [. A / x ]. -. ph \/ [. A / x ]. ph )
9		ralsns 3431	. . . . . 6 |- ( A e. _V -> ( A. x e. { A } -. ph <-> [. A / x ]. -. ph ) )
10	1, 9	ax-mp 7	. . . . 5 |- ( A. x e. { A } -. ph <-> [. A / x ]. -. ph )
11		ralsns 3431	. . . . . 6 |- ( A e. _V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )
12	1, 11	ax-mp 7	. . . . 5 |- ( A. x e. { A } ph <-> [. A / x ]. ph )
13	10, 12	orbi12i 713	. . . 4 |- ( ( A. x e. { A } -. ph \/ A. x e. { A } ph ) <-> ( [. A / x ]. -. ph \/ [. A / x ]. ph ) )
14	8, 13	mpbir 144	. . 3 |- ( A. x e. { A } -. ph \/ A. x e. { A } ph )
15		rabeq0 3274	. . . 4 |- ( { x e. { A } | ph } = (/) <-> A. x e. { A } -. ph )
16		eqcom 2083	. . . . 5 |- ( { x e. { A } | ph } = { A } <-> { A } = { x e. { A } | ph } )
17		rabid2 2530	. . . . 5 |- ( { A } = { x e. { A } | ph } <-> A. x e. { A } ph )
18	16, 17	bitri 182	. . . 4 |- ( { x e. { A } | ph } = { A } <-> A. x e. { A } ph )
19	15, 18	orbi12i 713	. . 3 |- ( ( { x e. { A } | ph } = (/) \/ { x e. { A } | ph } = { A } ) <-> ( A. x e. { A } -. ph \/ A. x e. { A } ph ) )
20	14, 19	mpbir 144	. 2 |- ( { x e. { A } | ph } = (/) \/ { x e. { A } | ph } = { A } )
21		eqeq1 2087	. . 3 |- ( M = { x e. { A } | ph } -> ( M = (/) <-> { x e. { A } | ph } = (/) ) )
22		eqeq1 2087	. . 3 |- ( M = { x e. { A } | ph } -> ( M = { A } <-> { x e. { A } | ph } = { A } ) )
23	21, 22	orbi12d 739	. 2 |- ( M = { x e. { A } | ph } -> ( ( M = (/) \/ M = { A } ) <-> ( { x e. { A } | ph } = (/) \/ { x e. { A } | ph } = { A } ) ) )
24	20, 23	mpbiri 166	1 |- ( M = { x e. { A } | ph } -> ( M = (/) \/ M = { A } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   \/ wo 661  DECID wdc 775   = wceq 1284   e. wcel 1433   A.wral 2348   {crab 2352   _Vcvv 2601   [.wsbc 2815   (/)c0 3251   {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-dc 776  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-nul 3252  df-sn 3404

This theorem is referenced by: (None)