Theorem exmidsssn 4125

Description: Excluded middle is equivalent to the biconditionalized version of sssnr 3680 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 3401	. . . . . . 7 |- (/) C_ { y }
2		sseq1 3120	. . . . . . 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 722	. . . . 5 |- ( (EXMID /\ x = (/) ) -> ( x = (/) \/ x = { y } ) )
7	4, 6	2thd 174	. . . 4 |- ( (EXMID /\ x = (/) ) -> ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
8		sssnm 3681	. . . . . 6 |- ( E. z z e. x -> ( x C_ { y } <-> x = { y } ) )
9		neq0r 3377	. . . . . . 7 |- ( E. z z e. x -> -. x = (/) )
10		biorf 733	. . . . . . 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 4124	. . . . . 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 1537	. . . 4 |- (EXMID -> ( x = (/) \/ E. z z e. x ) )
17	7, 13, 16	mpjaodan 787	. . 3 |- (EXMID -> ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
18	17	alrimivv 1847	. 2 |- (EXMID -> A. x A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
19		0ex 4055	. . . . . 6 |- (/) e. _V
20		sneq 3538	. . . . . . . 8 |- ( y = (/) -> { y } = { (/) } )
21	20	sseq2d 3127	. . . . . . 7 |- ( y = (/) -> ( x C_ { y } <-> x C_ { (/) } ) )
22	20	eqeq2d 2151	. . . . . . . 8 |- ( y = (/) -> ( x = { y } <-> x = { (/) } ) )
23	22	orbi2d 779	. . . . . . 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 2779	. . . . 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 1431	. . 3 |- ( A. x A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
28		exmid01 4121	. . 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 697   A.wal 1329   = wceq 1331   E.wex 1468   C_ wss 3071   (/)c0 3363   {csn 3527  EXMIDwem 4118

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098

This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-rab 2425  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-exmid 4119

This theorem is referenced by:  exmidsssnc  4126