Theorem exmidsssn 4201

Description: Excluded middle is equivalent to the biconditionalized version of sssnr 3753 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 3461	. . . . . . 7 |- (/) C_ { y }
2		sseq1 3178	. . . . . . 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 733	. . . . 5 |- ( (EXMID /\ x = (/) ) -> ( x = (/) \/ x = { y } ) )
7	4, 6	2thd 175	. . . 4 |- ( (EXMID /\ x = (/) ) -> ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
8		sssnm 3754	. . . . . 6 |- ( E. z z e. x -> ( x C_ { y } <-> x = { y } ) )
9		neq0r 3437	. . . . . . 7 |- ( E. z z e. x -> -. x = (/) )
10		biorf 744	. . . . . . 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 4200	. . . . . 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 1558	. . . 4 |- (EXMID -> ( x = (/) \/ E. z z e. x ) )
17	7, 13, 16	mpjaodan 798	. . 3 |- (EXMID -> ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
18	17	alrimivv 1875	. 2 |- (EXMID -> A. x A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
19		0ex 4129	. . . . . 6 |- (/) e. _V
20		sneq 3603	. . . . . . . 8 |- ( y = (/) -> { y } = { (/) } )
21	20	sseq2d 3185	. . . . . . 7 |- ( y = (/) -> ( x C_ { y } <-> x C_ { (/) } ) )
22	20	eqeq2d 2189	. . . . . . . 8 |- ( y = (/) -> ( x = { y } <-> x = { (/) } ) )
23	22	orbi2d 790	. . . . . . 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 2831	. . . . 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 1455	. . 3 |- ( A. x A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
28		exmid01 4197	. . 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 708   A.wal 1351   = wceq 1353   E.wex 1492   C_ wss 3129   (/)c0 3422   {csn 3592  EXMIDwem 4193

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-nul 4128  ax-pow 4173

This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-rab 2464  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-exmid 4194

This theorem is referenced by:  exmidsssnc  4202