Theorem exmidsssnc 4182

Description: Excluded middle in terms of subsets of a singleton. This is similar to exmid01 4177 but lets you choose any set as the element of the singleton rather than just (/). It is similar to exmidsssn 4181 but for a particular set B rather than all sets. (Contributed by Jim Kingdon, 29-Jul-2023.)

Assertion

Ref	Expression
exmidsssnc	|- ( B e. V -> (EXMID <-> A. x ( x C_ { B } -> ( x = (/) \/ x = { B } ) ) ) )

Distinct variable groups:   x, B   x, V

Proof of Theorem exmidsssnc

Dummy variables u w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		exmidsssn 4181	. . . 4 |- (EXMID <-> A. x A. u ( x C_ { u } <-> ( x = (/) \/ x = { u } ) ) )
2		sneq 3587	. . . . . . . 8 |- ( u = B -> { u } = { B } )
3	2	sseq2d 3172	. . . . . . 7 |- ( u = B -> ( x C_ { u } <-> x C_ { B } ) )
4	2	eqeq2d 2177	. . . . . . . 8 |- ( u = B -> ( x = { u } <-> x = { B } ) )
5	4	orbi2d 780	. . . . . . 7 |- ( u = B -> ( ( x = (/) \/ x = { u } ) <-> ( x = (/) \/ x = { B } ) ) )
6	3, 5	bibi12d 234	. . . . . 6 |- ( u = B -> ( ( x C_ { u } <-> ( x = (/) \/ x = { u } ) ) <-> ( x C_ { B } <-> ( x = (/) \/ x = { B } ) ) ) )
7	6	spcgv 2813	. . . . 5 |- ( B e. V -> ( A. u ( x C_ { u } <-> ( x = (/) \/ x = { u } ) ) -> ( x C_ { B } <-> ( x = (/) \/ x = { B } ) ) ) )
8	7	alimdv 1867	. . . 4 |- ( B e. V -> ( A. x A. u ( x C_ { u } <-> ( x = (/) \/ x = { u } ) ) -> A. x ( x C_ { B } <-> ( x = (/) \/ x = { B } ) ) ) )
9	1, 8	syl5bi 151	. . 3 |- ( B e. V -> (EXMID -> A. x ( x C_ { B } <-> ( x = (/) \/ x = { B } ) ) ) )
10		biimp 117	. . . 4 |- ( ( x C_ { B } <-> ( x = (/) \/ x = { B } ) ) -> ( x C_ { B } -> ( x = (/) \/ x = { B } ) ) )
11	10	alimi 1443	. . 3 |- ( A. x ( x C_ { B } <-> ( x = (/) \/ x = { B } ) ) -> A. x ( x C_ { B } -> ( x = (/) \/ x = { B } ) ) )
12	9, 11	syl6 33	. 2 |- ( B e. V -> (EXMID -> A. x ( x C_ { B } -> ( x = (/) \/ x = { B } ) ) ) )
13		ssrab2 3227	. . . . . . . . 9 |- { z e. { B } | (/) e. y } C_ { B }
14		snexg 4163	. . . . . . . . . 10 |- ( B e. V -> { B } e. _V )
15		rabexg 4125	. . . . . . . . . 10 |- ( { B } e. _V -> { z e. { B } | (/) e. y } e. _V )
16		sseq1 3165	. . . . . . . . . . . 12 |- ( x = { z e. { B } | (/) e. y } -> ( x C_ { B } <-> { z e. { B } | (/) e. y } C_ { B } ) )
17		eqeq1 2172	. . . . . . . . . . . . 13 |- ( x = { z e. { B } | (/) e. y } -> ( x = (/) <-> { z e. { B } | (/) e. y } = (/) ) )
18		eqeq1 2172	. . . . . . . . . . . . 13 |- ( x = { z e. { B } | (/) e. y } -> ( x = { B } <-> { z e. { B } | (/) e. y } = { B } ) )
19	17, 18	orbi12d 783	. . . . . . . . . . . 12 |- ( x = { z e. { B } | (/) e. y } -> ( ( x = (/) \/ x = { B } ) <-> ( { z e. { B } | (/) e. y } = (/) \/ { z e. { B } | (/) e. y } = { B } ) ) )
20	16, 19	imbi12d 233	. . . . . . . . . . 11 |- ( x = { z e. { B } | (/) e. y } -> ( ( x C_ { B } -> ( x = (/) \/ x = { B } ) ) <-> ( { z e. { B } | (/) e. y } C_ { B } -> ( { z e. { B } | (/) e. y } = (/) \/ { z e. { B } | (/) e. y } = { B } ) ) ) )
21	20	spcgv 2813	. . . . . . . . . 10 |- ( { z e. { B } | (/) e. y } e. _V -> ( A. x ( x C_ { B } -> ( x = (/) \/ x = { B } ) ) -> ( { z e. { B } | (/) e. y } C_ { B } -> ( { z e. { B } | (/) e. y } = (/) \/ { z e. { B } | (/) e. y } = { B } ) ) ) )
22	14, 15, 21	3syl 17	. . . . . . . . 9 |- ( B e. V -> ( A. x ( x C_ { B } -> ( x = (/) \/ x = { B } ) ) -> ( { z e. { B } | (/) e. y } C_ { B } -> ( { z e. { B } | (/) e. y } = (/) \/ { z e. { B } | (/) e. y } = { B } ) ) ) )
23	13, 22	mpii 44	. . . . . . . 8 |- ( B e. V -> ( A. x ( x C_ { B } -> ( x = (/) \/ x = { B } ) ) -> ( { z e. { B } | (/) e. y } = (/) \/ { z e. { B } | (/) e. y } = { B } ) ) )
24		rabeq0 3438	. . . . . . . . . . 11 |- ( { z e. { B } | (/) e. y } = (/) <-> A. z e. { B } -. (/) e. y )
25		snmg 3694	. . . . . . . . . . . 12 |- ( B e. V -> E. w w e. { B } )
26		r19.3rmv 3499	. . . . . . . . . . . 12 |- ( E. w w e. { B } -> ( -. (/) e. y <-> A. z e. { B } -. (/) e. y ) )
27	25, 26	syl 14	. . . . . . . . . . 11 |- ( B e. V -> ( -. (/) e. y <-> A. z e. { B } -. (/) e. y ) )
28	24, 27	bitr4id 198	. . . . . . . . . 10 |- ( B e. V -> ( { z e. { B } | (/) e. y } = (/) <-> -. (/) e. y ) )
29	28	biimpd 143	. . . . . . . . 9 |- ( B e. V -> ( { z e. { B } | (/) e. y } = (/) -> -. (/) e. y ) )
30		snidg 3605	. . . . . . . . . . . . 13 |- ( B e. V -> B e. { B } )
31	30	adantr 274	. . . . . . . . . . . 12 |- ( ( B e. V /\ { z e. { B } | (/) e. y } = { B } ) -> B e. { B } )
32		simpr 109	. . . . . . . . . . . 12 |- ( ( B e. V /\ { z e. { B } | (/) e. y } = { B } ) -> { z e. { B } | (/) e. y } = { B } )
33	31, 32	eleqtrrd 2246	. . . . . . . . . . 11 |- ( ( B e. V /\ { z e. { B } | (/) e. y } = { B } ) -> B e. { z e. { B } | (/) e. y } )
34		biidd 171	. . . . . . . . . . . . 13 |- ( z = B -> ( (/) e. y <-> (/) e. y ) )
35	34	elrab 2882	. . . . . . . . . . . 12 |- ( B e. { z e. { B } | (/) e. y } <-> ( B e. { B } /\ (/) e. y ) )
36	35	simprbi 273	. . . . . . . . . . 11 |- ( B e. { z e. { B } | (/) e. y } -> (/) e. y )
37	33, 36	syl 14	. . . . . . . . . 10 |- ( ( B e. V /\ { z e. { B } | (/) e. y } = { B } ) -> (/) e. y )
38	37	ex 114	. . . . . . . . 9 |- ( B e. V -> ( { z e. { B } | (/) e. y } = { B } -> (/) e. y ) )
39	29, 38	orim12d 776	. . . . . . . 8 |- ( B e. V -> ( ( { z e. { B } | (/) e. y } = (/) \/ { z e. { B } | (/) e. y } = { B } ) -> ( -. (/) e. y \/ (/) e. y ) ) )
40	23, 39	syld 45	. . . . . . 7 |- ( B e. V -> ( A. x ( x C_ { B } -> ( x = (/) \/ x = { B } ) ) -> ( -. (/) e. y \/ (/) e. y ) ) )
41		orcom 718	. . . . . . 7 |- ( ( -. (/) e. y \/ (/) e. y ) <-> ( (/) e. y \/ -. (/) e. y ) )
42	40, 41	syl6ib 160	. . . . . 6 |- ( B e. V -> ( A. x ( x C_ { B } -> ( x = (/) \/ x = { B } ) ) -> ( (/) e. y \/ -. (/) e. y ) ) )
43		df-dc 825	. . . . . 6 |- (DECID (/) e. y <-> ( (/) e. y \/ -. (/) e. y ) )
44	42, 43	syl6ibr 161	. . . . 5 |- ( B e. V -> ( A. x ( x C_ { B } -> ( x = (/) \/ x = { B } ) ) -> DECID (/) e. y ) )
45	44	a1dd 48	. . . 4 |- ( B e. V -> ( A. x ( x C_ { B } -> ( x = (/) \/ x = { B } ) ) -> ( y C_ { (/) } -> DECID (/) e. y ) ) )
46	45	alrimdv 1864	. . 3 |- ( B e. V -> ( A. x ( x C_ { B } -> ( x = (/) \/ x = { B } ) ) -> A. y ( y C_ { (/) } -> DECID (/) e. y ) ) )
47		df-exmid 4174	. . 3 |- (EXMID <-> A. y ( y C_ { (/) } -> DECID (/) e. y ) )
48	46, 47	syl6ibr 161	. 2 |- ( B e. V -> ( A. x ( x C_ { B } -> ( x = (/) \/ x = { B } ) ) -> EXMID ) )
49	12, 48	impbid 128	1 |- ( B e. V -> (EXMID <-> A. x ( x C_ { B } -> ( x = (/) \/ x = { B } ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 824   A.wal 1341   = wceq 1343   E.wex 1480   e. wcel 2136   A.wral 2444   {crab 2448   _Vcvv 2726   C_ wss 3116   (/)c0 3409   {csn 3576  EXMIDwem 4173

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rab 2453  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-exmid 4174

This theorem is referenced by:  exmidunben  12359