Theorem exmidsssn 4262

Description: Excluded middle is equivalent to the biconditionalized version of sssnr 3807 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 3507	. . . . . . 7 |- (/) C_ { y }
2		sseq1 3224	. . . . . . 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 735	. . . . 5 |- ( (EXMID /\ x = (/) ) -> ( x = (/) \/ x = { y } ) )
7	4, 6	2thd 175	. . . 4 |- ( (EXMID /\ x = (/) ) -> ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
8		sssnm 3808	. . . . . 6 |- ( E. z z e. x -> ( x C_ { y } <-> x = { y } ) )
9		neq0r 3483	. . . . . . 7 |- ( E. z z e. x -> -. x = (/) )
10		biorf 746	. . . . . . 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 4261	. . . . . 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 1582	. . . 4 |- (EXMID -> ( x = (/) \/ E. z z e. x ) )
17	7, 13, 16	mpjaodan 800	. . 3 |- (EXMID -> ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
18	17	alrimivv 1899	. 2 |- (EXMID -> A. x A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
19		0ex 4187	. . . . . 6 |- (/) e. _V
20		sneq 3654	. . . . . . . 8 |- ( y = (/) -> { y } = { (/) } )
21	20	sseq2d 3231	. . . . . . 7 |- ( y = (/) -> ( x C_ { y } <-> x C_ { (/) } ) )
22	20	eqeq2d 2219	. . . . . . . 8 |- ( y = (/) -> ( x = { y } <-> x = { (/) } ) )
23	22	orbi2d 792	. . . . . . 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 2874	. . . . 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 1479	. . 3 |- ( A. x A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
28		exmid01 4258	. . 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 710   A.wal 1371   = wceq 1373   E.wex 1516   C_ wss 3174   (/)c0 3468   {csn 3643  EXMIDwem 4254

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-rab 2495  df-v 2778  df-dif 3176  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-exmid 4255

This theorem is referenced by:  exmidsssnc  4263