Theorem exmidsssn 4181

Description: Excluded middle is equivalent to the biconditionalized version of sssnr 3733 for sets. (Contributed by Jim Kingdon, 5-Mar-2023.)

Assertion

Ref	Expression
exmidsssn	|- (EXMID <-> A. x A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )

Distinct variable group:   x, y

Proof of Theorem exmidsssn

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ss 3447	. . . . . . 7 |- (/) C_ { y }
2		sseq1 3165	. . . . . . 7 |- ( x = (/) -> ( x C_ { y } <-> (/) C_ { y } ) )
3	1, 2	mpbiri 167	. . . . . 6 |- ( x = (/) -> x C_ { y } )
4	3	adantl 275	. . . . 5 |- ( (EXMID /\ x = (/) ) -> x C_ { y } )
5		simpr 109	. . . . . 6 |- ( (EXMID /\ x = (/) ) -> x = (/) )
6	5	orcd 723	. . . . 5 |- ( (EXMID /\ x = (/) ) -> ( x = (/) \/ x = { y } ) )
7	4, 6	2thd 174	. . . 4 |- ( (EXMID /\ x = (/) ) -> ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
8		sssnm 3734	. . . . . 6 |- ( E. z z e. x -> ( x C_ { y } <-> x = { y } ) )
9		neq0r 3423	. . . . . . 7 |- ( E. z z e. x -> -. x = (/) )
10		biorf 734	. . . . . . 7 |- ( -. x = (/) -> ( x = { y } <-> ( x = (/) \/ x = { y } ) ) )
11	9, 10	syl 14	. . . . . 6 |- ( E. z z e. x -> ( x = { y } <-> ( x = (/) \/ x = { y } ) ) )
12	8, 11	bitrd 187	. . . . 5 |- ( E. z z e. x -> ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
13	12	adantl 275	. . . 4 |- ( (EXMID /\ E. z z e. x ) -> ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
14		exmidn0m 4180	. . . . . 6 |- (EXMID <-> A. x ( x = (/) \/ E. z z e. x ) )
15	14	biimpi 119	. . . . 5 |- (EXMID -> A. x ( x = (/) \/ E. z z e. x ) )
16	15	19.21bi 1546	. . . 4 |- (EXMID -> ( x = (/) \/ E. z z e. x ) )
17	7, 13, 16	mpjaodan 788	. . 3 |- (EXMID -> ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
18	17	alrimivv 1863	. 2 |- (EXMID -> A. x A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
19		0ex 4109	. . . . . 6 |- (/) e. _V
20		sneq 3587	. . . . . . . 8 |- ( y = (/) -> { y } = { (/) } )
21	20	sseq2d 3172	. . . . . . 7 |- ( y = (/) -> ( x C_ { y } <-> x C_ { (/) } ) )
22	20	eqeq2d 2177	. . . . . . . 8 |- ( y = (/) -> ( x = { y } <-> x = { (/) } ) )
23	22	orbi2d 780	. . . . . . 7 |- ( y = (/) -> ( ( x = (/) \/ x = { y } ) <-> ( x = (/) \/ x = { (/) } ) ) )
24	21, 23	bibi12d 234	. . . . . 6 |- ( y = (/) -> ( ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) <-> ( x C_ { (/) } <-> ( x = (/) \/ x = { (/) } ) ) ) )
25	19, 24	spcv 2820	. . . . 5 |- ( A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) -> ( x C_ { (/) } <-> ( x = (/) \/ x = { (/) } ) ) )
26	25	biimpd 143	. . . 4 |- ( A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
27	26	alimi 1443	. . 3 |- ( A. x A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
28		exmid01 4177	. . 3 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
29	27, 28	sylibr 133	. 2 |- ( A. x A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) -> EXMID )
30	18, 29	impbii 125	1 |- (EXMID <-> A. x A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   A.wal 1341   = wceq 1343   E.wex 1480   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-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:  exmidsssnc  4182