Theorem exmid01 4129

Description: Excluded middle is equivalent to saying any subset of { (/) } is either (/) or { (/) }. (Contributed by BJ and Jim Kingdon, 18-Jun-2022.)

Assertion

Ref	Expression
exmid01	|- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )

Proof of Theorem exmid01

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-exmid 4127	. 2 |- (EXMID <-> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
2		df-dc 821	. . . . 5 |- (DECID (/) e. x <-> ( (/) e. x \/ -. (/) e. x ) )
3		orcom 718	. . . . . 6 |- ( ( (/) e. x \/ -. (/) e. x ) <-> ( -. (/) e. x \/ (/) e. x ) )
4		simpll 519	. . . . . . . . . . . . . 14 |- ( ( ( x C_ { (/) } /\ -. (/) e. x ) /\ y e. x ) -> x C_ { (/) } )
5		simpr 109	. . . . . . . . . . . . . 14 |- ( ( ( x C_ { (/) } /\ -. (/) e. x ) /\ y e. x ) -> y e. x )
6	4, 5	sseldd 3103	. . . . . . . . . . . . 13 |- ( ( ( x C_ { (/) } /\ -. (/) e. x ) /\ y e. x ) -> y e. { (/) } )
7		velsn 3549	. . . . . . . . . . . . 13 |- ( y e. { (/) } <-> y = (/) )
8	6, 7	sylib 121	. . . . . . . . . . . 12 |- ( ( ( x C_ { (/) } /\ -. (/) e. x ) /\ y e. x ) -> y = (/) )
9	8, 5	eqeltrrd 2218	. . . . . . . . . . 11 |- ( ( ( x C_ { (/) } /\ -. (/) e. x ) /\ y e. x ) -> (/) e. x )
10		simplr 520	. . . . . . . . . . 11 |- ( ( ( x C_ { (/) } /\ -. (/) e. x ) /\ y e. x ) -> -. (/) e. x )
11	9, 10	pm2.65da 651	. . . . . . . . . 10 |- ( ( x C_ { (/) } /\ -. (/) e. x ) -> -. y e. x )
12	11	eq0rdv 3412	. . . . . . . . 9 |- ( ( x C_ { (/) } /\ -. (/) e. x ) -> x = (/) )
13	12	ex 114	. . . . . . . 8 |- ( x C_ { (/) } -> ( -. (/) e. x -> x = (/) ) )
14		noel 3372	. . . . . . . . 9 |- -. (/) e. (/)
15		eleq2 2204	. . . . . . . . 9 |- ( x = (/) -> ( (/) e. x <-> (/) e. (/) ) )
16	14, 15	mtbiri 665	. . . . . . . 8 |- ( x = (/) -> -. (/) e. x )
17	13, 16	impbid1 141	. . . . . . 7 |- ( x C_ { (/) } -> ( -. (/) e. x <-> x = (/) ) )
18		elex2 2705	. . . . . . . . . 10 |- ( (/) e. x -> E. z z e. x )
19		sssnm 3689	. . . . . . . . . 10 |- ( E. z z e. x -> ( x C_ { (/) } <-> x = { (/) } ) )
20	18, 19	syl 14	. . . . . . . . 9 |- ( (/) e. x -> ( x C_ { (/) } <-> x = { (/) } ) )
21	20	biimpcd 158	. . . . . . . 8 |- ( x C_ { (/) } -> ( (/) e. x -> x = { (/) } ) )
22		0ex 4063	. . . . . . . . . 10 |- (/) e. _V
23	22	snid 3563	. . . . . . . . 9 |- (/) e. { (/) }
24		eleq2 2204	. . . . . . . . 9 |- ( x = { (/) } -> ( (/) e. x <-> (/) e. { (/) } ) )
25	23, 24	mpbiri 167	. . . . . . . 8 |- ( x = { (/) } -> (/) e. x )
26	21, 25	impbid1 141	. . . . . . 7 |- ( x C_ { (/) } -> ( (/) e. x <-> x = { (/) } ) )
27	17, 26	orbi12d 783	. . . . . 6 |- ( x C_ { (/) } -> ( ( -. (/) e. x \/ (/) e. x ) <-> ( x = (/) \/ x = { (/) } ) ) )
28	3, 27	syl5bb 191	. . . . 5 |- ( x C_ { (/) } -> ( ( (/) e. x \/ -. (/) e. x ) <-> ( x = (/) \/ x = { (/) } ) ) )
29	2, 28	syl5bb 191	. . . 4 |- ( x C_ { (/) } -> (DECID (/) e. x <-> ( x = (/) \/ x = { (/) } ) ) )
30	29	pm5.74i 179	. . 3 |- ( ( x C_ { (/) } -> DECID (/) e. x ) <-> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
31	30	albii 1447	. 2 |- ( A. x ( x C_ { (/) } -> DECID (/) e. x ) <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
32	1, 31	bitri 183	1 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 820   A.wal 1330   = wceq 1332   E.wex 1469   e. wcel 1481   C_ wss 3076   (/)c0 3368   {csn 3532  EXMIDwem 4126

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-nul 4062

This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089  df-nul 3369  df-sn 3538  df-exmid 4127

This theorem is referenced by:  exmid1dc  4131  exmidn0m  4132  exmidsssn  4133  exmidpw  6810  exmidomni  7022  ss1oel2o  13360  exmidsbthrlem  13392  sbthom  13396