Theorem rabrsndc 3557

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 805	. . . . . . . 8 |- (DECID ph -> ( -. ph \/ ph ) )
4	2, 3	ax-mp 7	. . . . . . 7 |- ( -. ph \/ ph )
5	4	sbcth 2891	. . . . . 6 |- ( A e. _V -> [. A / x ]. ( -. ph \/ ph ) )
6	1, 5	ax-mp 7	. . . . 5 |- [. A / x ]. ( -. ph \/ ph )
7		sbcor 2921	. . . . 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 3528	. . . . . 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 3528	. . . . . 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 736	. . . 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 3358	. . . 4 |- ( { x e. { A } | ph } = (/) <-> A. x e. { A } -. ph )
16		eqcom 2117	. . . . 5 |- ( { x e. { A } | ph } = { A } <-> { A } = { x e. { A } | ph } )
17		rabid2 2581	. . . . 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 736	. . 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 2121	. . 3 |- ( M = { x e. { A } | ph } -> ( M = (/) <-> { x e. { A } | ph } = (/) ) )
22		eqeq1 2121	. . 3 |- ( M = { x e. { A } | ph } -> ( M = { A } <-> { x e. { A } | ph } = { A } ) )
23	21, 22	orbi12d 765	. 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 680  DECID wdc 802   = wceq 1314   e. wcel 1463   A.wral 2390   {crab 2394   _Vcvv 2657   [.wsbc 2878   (/)c0 3329   {csn 3493

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-dc 803  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-nul 3330  df-sn 3499

This theorem is referenced by: (None)