Theorem exmid01 4199

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 4196	. 2 |- (EXMID <-> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
2		df-dc 835	. . . . 5 |- (DECID (/) e. x <-> ( (/) e. x \/ -. (/) e. x ) )
3		orcom 728	. . . . . 6 |- ( ( (/) e. x \/ -. (/) e. x ) <-> ( -. (/) e. x \/ (/) e. x ) )
4		simpll 527	. . . . . . . . . . . . . 14 |- ( ( ( x C_ { (/) } /\ -. (/) e. x ) /\ y e. x ) -> x C_ { (/) } )
5		simpr 110	. . . . . . . . . . . . . 14 |- ( ( ( x C_ { (/) } /\ -. (/) e. x ) /\ y e. x ) -> y e. x )
6	4, 5	sseldd 3157	. . . . . . . . . . . . 13 |- ( ( ( x C_ { (/) } /\ -. (/) e. x ) /\ y e. x ) -> y e. { (/) } )
7		velsn 3610	. . . . . . . . . . . . 13 |- ( y e. { (/) } <-> y = (/) )
8	6, 7	sylib 122	. . . . . . . . . . . 12 |- ( ( ( x C_ { (/) } /\ -. (/) e. x ) /\ y e. x ) -> y = (/) )
9	8, 5	eqeltrrd 2255	. . . . . . . . . . 11 |- ( ( ( x C_ { (/) } /\ -. (/) e. x ) /\ y e. x ) -> (/) e. x )
10		simplr 528	. . . . . . . . . . 11 |- ( ( ( x C_ { (/) } /\ -. (/) e. x ) /\ y e. x ) -> -. (/) e. x )
11	9, 10	pm2.65da 661	. . . . . . . . . 10 |- ( ( x C_ { (/) } /\ -. (/) e. x ) -> -. y e. x )
12	11	eq0rdv 3468	. . . . . . . . 9 |- ( ( x C_ { (/) } /\ -. (/) e. x ) -> x = (/) )
13	12	ex 115	. . . . . . . 8 |- ( x C_ { (/) } -> ( -. (/) e. x -> x = (/) ) )
14		noel 3427	. . . . . . . . 9 |- -. (/) e. (/)
15		eleq2 2241	. . . . . . . . 9 |- ( x = (/) -> ( (/) e. x <-> (/) e. (/) ) )
16	14, 15	mtbiri 675	. . . . . . . 8 |- ( x = (/) -> -. (/) e. x )
17	13, 16	impbid1 142	. . . . . . 7 |- ( x C_ { (/) } -> ( -. (/) e. x <-> x = (/) ) )
18		ss1o0el1 4198	. . . . . . 7 |- ( x C_ { (/) } -> ( (/) e. x <-> x = { (/) } ) )
19	17, 18	orbi12d 793	. . . . . 6 |- ( x C_ { (/) } -> ( ( -. (/) e. x \/ (/) e. x ) <-> ( x = (/) \/ x = { (/) } ) ) )
20	3, 19	bitrid 192	. . . . 5 |- ( x C_ { (/) } -> ( ( (/) e. x \/ -. (/) e. x ) <-> ( x = (/) \/ x = { (/) } ) ) )
21	2, 20	bitrid 192	. . . 4 |- ( x C_ { (/) } -> (DECID (/) e. x <-> ( x = (/) \/ x = { (/) } ) ) )
22	21	pm5.74i 180	. . 3 |- ( ( x C_ { (/) } -> DECID (/) e. x ) <-> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
23	22	albii 1470	. 2 |- ( A. x ( x C_ { (/) } -> DECID (/) e. x ) <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
24	1, 23	bitri 184	1 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708  DECID wdc 834   A.wal 1351   = wceq 1353   e. wcel 2148   C_ wss 3130   (/)c0 3423   {csn 3593  EXMIDwem 4195

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-ext 2159  ax-nul 4130

This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-dif 3132  df-in 3136  df-ss 3143  df-nul 3424  df-sn 3599  df-exmid 4196

This theorem is referenced by:  exmid1dc  4201  exmidn0m  4202  exmidsssn  4203  exmidpw  6908  exmidpweq  6909  exmidomni  7140  ss1oel2o  14746  exmidsbthrlem  14773  sbthom  14777