Theorem rabrsndc 3438

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 745	. . . . . . . 8 |- (DECID ph -> ( -. ph \/ ph ) )
4	2, 3	ax-mp 7	. . . . . . 7 |- ( -. ph \/ ph )
5	4	sbcth 2777	. . . . . 6 |- ( A e. _V -> [. A / x ]. ( -. ph \/ ph ) )
6	1, 5	ax-mp 7	. . . . 5 |- [. A / x ]. ( -. ph \/ ph )
7		sbcor 2807	. . . . 5 |- ( [. A / x ]. ( -. ph \/ ph ) <-> ( [. A / x ]. -. ph \/ [. A / x ]. ph ) )
8	6, 7	mpbi 133	. . . 4 |- ( [. A / x ]. -. ph \/ [. A / x ]. ph )
9		ralsns 3408	. . . . . 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 3408	. . . . . 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 681	. . . 4 |- ( ( A. x e. { A } -. ph \/ A. x e. { A } ph ) <-> ( [. A / x ]. -. ph \/ [. A / x ]. ph ) )
14	8, 13	mpbir 134	. . 3 |- ( A. x e. { A } -. ph \/ A. x e. { A } ph )
15		rabeq0 3247	. . . 4 |- ( { x e. { A } | ph } = (/) <-> A. x e. { A } -. ph )
16		eqcom 2042	. . . . 5 |- ( { x e. { A } | ph } = { A } <-> { A } = { x e. { A } | ph } )
17		rabid2 2486	. . . . 5 |- ( { A } = { x e. { A } | ph } <-> A. x e. { A } ph )
18	16, 17	bitri 173	. . . 4 |- ( { x e. { A } | ph } = { A } <-> A. x e. { A } ph )
19	15, 18	orbi12i 681	. . 3 |- ( ( { x e. { A } | ph } = (/) \/ { x e. { A } | ph } = { A } ) <-> ( A. x e. { A } -. ph \/ A. x e. { A } ph ) )
20	14, 19	mpbir 134	. 2 |- ( { x e. { A } | ph } = (/) \/ { x e. { A } | ph } = { A } )
21		eqeq1 2046	. . 3 |- ( M = { x e. { A } | ph } -> ( M = (/) <-> { x e. { A } | ph } = (/) ) )
22		eqeq1 2046	. . 3 |- ( M = { x e. { A } | ph } -> ( M = { A } <-> { x e. { A } | ph } = { A } ) )
23	21, 22	orbi12d 707	. 2 |- ( M = { x e. { A } | ph } -> ( ( M = (/) \/ M = { A } ) <-> ( { x e. { A } | ph } = (/) \/ { x e. { A } | ph } = { A } ) ) )
24	20, 23	mpbiri 157	1 |- ( M = { x e. { A } | ph } -> ( M = (/) \/ M = { A } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98   \/ wo 629  DECID wdc 742   = wceq 1243   e. wcel 1393   A.wral 2306   {crab 2310   _Vcvv 2557   [.wsbc 2764   (/)c0 3224   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-dc 743  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-nul 3225  df-sn 3381

This theorem is referenced by: (None)