Theorem exmid01 4177

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 variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-exmid 4174	. 2 |- (EXMID <-> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
2		df-dc 825	. . . . 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 3143	. . . . . . . . . . . . 13 |- ( ( ( x C_ { (/) } /\ -. (/) e. x ) /\ y e. x ) -> y e. { (/) } )
7		velsn 3593	. . . . . . . . . . . . 13 |- ( y e. { (/) } <-> y = (/) )
8	6, 7	sylib 121	. . . . . . . . . . . 12 |- ( ( ( x C_ { (/) } /\ -. (/) e. x ) /\ y e. x ) -> y = (/) )
9	8, 5	eqeltrrd 2244	. . . . . . . . . . 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 3453	. . . . . . . . 9 |- ( ( x C_ { (/) } /\ -. (/) e. x ) -> x = (/) )
13	12	ex 114	. . . . . . . 8 |- ( x C_ { (/) } -> ( -. (/) e. x -> x = (/) ) )
14		noel 3413	. . . . . . . . 9 |- -. (/) e. (/)
15		eleq2 2230	. . . . . . . . 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		ss1o0el1 4176	. . . . . . 7 |- ( x C_ { (/) } -> ( (/) e. x <-> x = { (/) } ) )
19	17, 18	orbi12d 783	. . . . . 6 |- ( x C_ { (/) } -> ( ( -. (/) e. x \/ (/) e. x ) <-> ( x = (/) \/ x = { (/) } ) ) )
20	3, 19	syl5bb 191	. . . . 5 |- ( x C_ { (/) } -> ( ( (/) e. x \/ -. (/) e. x ) <-> ( x = (/) \/ x = { (/) } ) ) )
21	2, 20	syl5bb 191	. . . 4 |- ( x C_ { (/) } -> (DECID (/) e. x <-> ( x = (/) \/ x = { (/) } ) ) )
22	21	pm5.74i 179	. . 3 |- ( ( x C_ { (/) } -> DECID (/) e. x ) <-> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
23	22	albii 1458	. 2 |- ( A. x ( x C_ { (/) } -> DECID (/) e. x ) <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
24	1, 23	bitri 183	1 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 824   A.wal 1341   = wceq 1343   e. wcel 2136   C_ wss 3116   (/)c0 3409   {csn 3576  EXMIDwem 4173

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-nul 4108

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410  df-sn 3582  df-exmid 4174

This theorem is referenced by:  exmid1dc  4179  exmidn0m  4180  exmidsssn  4181  exmidpw  6874  exmidpweq  6875  exmidomni  7106  ss1oel2o  13873  exmidsbthrlem  13901  sbthom  13905