Theorem rabrsndc 3627

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 823	. . . . . . . 8 |- (DECID ph -> ( -. ph \/ ph ) )
4	2, 3	ax-mp 5	. . . . . . 7 |- ( -. ph \/ ph )
5	4	sbcth 2950	. . . . . 6 |- ( A e. _V -> [. A / x ]. ( -. ph \/ ph ) )
6	1, 5	ax-mp 5	. . . . 5 |- [. A / x ]. ( -. ph \/ ph )
7		sbcor 2981	. . . . 5 |- ( [. A / x ]. ( -. ph \/ ph ) <-> ( [. A / x ]. -. ph \/ [. A / x ]. ph ) )
8	6, 7	mpbi 144	. . . 4 |- ( [. A / x ]. -. ph \/ [. A / x ]. ph )
9		ralsns 3597	. . . . . 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 3597	. . . . . 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 754	. . . 4 |- ( ( A. x e. { A } -. ph \/ A. x e. { A } ph ) <-> ( [. A / x ]. -. ph \/ [. A / x ]. ph ) )
14	8, 13	mpbir 145	. . 3 |- ( A. x e. { A } -. ph \/ A. x e. { A } ph )
15		rabeq0 3423	. . . 4 |- ( { x e. { A } | ph } = (/) <-> A. x e. { A } -. ph )
16		eqcom 2159	. . . . 5 |- ( { x e. { A } | ph } = { A } <-> { A } = { x e. { A } | ph } )
17		rabid2 2633	. . . . 5 |- ( { A } = { x e. { A } | ph } <-> A. x e. { A } ph )
18	16, 17	bitri 183	. . . 4 |- ( { x e. { A } | ph } = { A } <-> A. x e. { A } ph )
19	15, 18	orbi12i 754	. . 3 |- ( ( { x e. { A } | ph } = (/) \/ { x e. { A } | ph } = { A } ) <-> ( A. x e. { A } -. ph \/ A. x e. { A } ph ) )
20	14, 19	mpbir 145	. 2 |- ( { x e. { A } | ph } = (/) \/ { x e. { A } | ph } = { A } )
21		eqeq1 2164	. . 3 |- ( M = { x e. { A } | ph } -> ( M = (/) <-> { x e. { A } | ph } = (/) ) )
22		eqeq1 2164	. . 3 |- ( M = { x e. { A } | ph } -> ( M = { A } <-> { x e. { A } | ph } = { A } ) )
23	21, 22	orbi12d 783	. 2 |- ( M = { x e. { A } | ph } -> ( ( M = (/) \/ M = { A } ) <-> ( { x e. { A } | ph } = (/) \/ { x e. { A } | ph } = { A } ) ) )
24	20, 23	mpbiri 167	1 |- ( M = { x e. { A } | ph } -> ( M = (/) \/ M = { A } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   \/ wo 698  DECID wdc 820   = wceq 1335   e. wcel 2128   A.wral 2435   {crab 2439   _Vcvv 2712   [.wsbc 2937   (/)c0 3394   {csn 3560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-nul 3395  df-sn 3566

This theorem is referenced by: (None)