Theorem exmidsssn 4231

Description: Excluded middle is equivalent to the biconditionalized version of sssnr 3779 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 3485	. . . . . . 7 |- (/) C_ { y }
2		sseq1 3202	. . . . . . 7 |- ( x = (/) -> ( x C_ { y } <-> (/) C_ { y } ) )
3	1, 2	mpbiri 168	. . . . . 6 |- ( x = (/) -> x C_ { y } )
4	3	adantl 277	. . . . 5 |- ( (EXMID /\ x = (/) ) -> x C_ { y } )
5		simpr 110	. . . . . 6 |- ( (EXMID /\ x = (/) ) -> x = (/) )
6	5	orcd 734	. . . . 5 |- ( (EXMID /\ x = (/) ) -> ( x = (/) \/ x = { y } ) )
7	4, 6	2thd 175	. . . 4 |- ( (EXMID /\ x = (/) ) -> ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
8		sssnm 3780	. . . . . 6 |- ( E. z z e. x -> ( x C_ { y } <-> x = { y } ) )
9		neq0r 3461	. . . . . . 7 |- ( E. z z e. x -> -. x = (/) )
10		biorf 745	. . . . . . 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 188	. . . . 5 |- ( E. z z e. x -> ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
13	12	adantl 277	. . . 4 |- ( (EXMID /\ E. z z e. x ) -> ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
14		exmidn0m 4230	. . . . . 6 |- (EXMID <-> A. x ( x = (/) \/ E. z z e. x ) )
15	14	biimpi 120	. . . . 5 |- (EXMID -> A. x ( x = (/) \/ E. z z e. x ) )
16	15	19.21bi 1569	. . . 4 |- (EXMID -> ( x = (/) \/ E. z z e. x ) )
17	7, 13, 16	mpjaodan 799	. . 3 |- (EXMID -> ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
18	17	alrimivv 1886	. 2 |- (EXMID -> A. x A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
19		0ex 4156	. . . . . 6 |- (/) e. _V
20		sneq 3629	. . . . . . . 8 |- ( y = (/) -> { y } = { (/) } )
21	20	sseq2d 3209	. . . . . . 7 |- ( y = (/) -> ( x C_ { y } <-> x C_ { (/) } ) )
22	20	eqeq2d 2205	. . . . . . . 8 |- ( y = (/) -> ( x = { y } <-> x = { (/) } ) )
23	22	orbi2d 791	. . . . . . 7 |- ( y = (/) -> ( ( x = (/) \/ x = { y } ) <-> ( x = (/) \/ x = { (/) } ) ) )
24	21, 23	bibi12d 235	. . . . . 6 |- ( y = (/) -> ( ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) <-> ( x C_ { (/) } <-> ( x = (/) \/ x = { (/) } ) ) ) )
25	19, 24	spcv 2854	. . . . 5 |- ( A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) -> ( x C_ { (/) } <-> ( x = (/) \/ x = { (/) } ) ) )
26	25	biimpd 144	. . . 4 |- ( A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
27	26	alimi 1466	. . 3 |- ( A. x A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
28		exmid01 4227	. . 3 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
29	27, 28	sylibr 134	. 2 |- ( A. x A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) -> EXMID )
30	18, 29	impbii 126	1 |- (EXMID <-> A. x A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   A.wal 1362   = wceq 1364   E.wex 1503   C_ wss 3153   (/)c0 3446   {csn 3618  EXMIDwem 4223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-rab 2481  df-v 2762  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-exmid 4224

This theorem is referenced by:  exmidsssnc  4232