Theorem exmid1dc 4179

Description: A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4172 or ordtriexmid 4498. In this context x = { (/) } can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.)

Hypothesis

Ref	Expression
exmid1dc.x	|- ( ( ph /\ x C_ { (/) } ) -> DECID x = { (/) } )

Assertion

Ref	Expression
exmid1dc	|- ( ph -> EXMID )

Distinct variable group:   ph, x

Proof of Theorem exmid1dc

Step	Hyp	Ref	Expression
1		exmid1dc.x	. . . . . . 7 |- ( ( ph /\ x C_ { (/) } ) -> DECID x = { (/) } )
2		exmiddc 826	. . . . . . 7 |- (DECID x = { (/) } -> ( x = { (/) } \/ -. x = { (/) } ) )
3	1, 2	syl 14	. . . . . 6 |- ( ( ph /\ x C_ { (/) } ) -> ( x = { (/) } \/ -. x = { (/) } ) )
4		df-ne 2337	. . . . . . . . 9 |- ( x =/= { (/) } <-> -. x = { (/) } )
5		pwntru 4178	. . . . . . . . . 10 |- ( ( x C_ { (/) } /\ x =/= { (/) } ) -> x = (/) )
6	5	ex 114	. . . . . . . . 9 |- ( x C_ { (/) } -> ( x =/= { (/) } -> x = (/) ) )
7	4, 6	syl5bir 152	. . . . . . . 8 |- ( x C_ { (/) } -> ( -. x = { (/) } -> x = (/) ) )
8	7	orim2d 778	. . . . . . 7 |- ( x C_ { (/) } -> ( ( x = { (/) } \/ -. x = { (/) } ) -> ( x = { (/) } \/ x = (/) ) ) )
9	8	adantl 275	. . . . . 6 |- ( ( ph /\ x C_ { (/) } ) -> ( ( x = { (/) } \/ -. x = { (/) } ) -> ( x = { (/) } \/ x = (/) ) ) )
10	3, 9	mpd 13	. . . . 5 |- ( ( ph /\ x C_ { (/) } ) -> ( x = { (/) } \/ x = (/) ) )
11	10	orcomd 719	. . . 4 |- ( ( ph /\ x C_ { (/) } ) -> ( x = (/) \/ x = { (/) } ) )
12	11	ex 114	. . 3 |- ( ph -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
13	12	alrimiv 1862	. 2 |- ( ph -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
14		exmid01 4177	. 2 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
15	13, 14	sylibr 133	1 |- ( ph -> EXMID )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698  DECID wdc 824   A.wal 1341   = wceq 1343   =/= wne 2336   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-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  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:  pw1fin  6876  exmidonfin  7150  exmidaclem  7164  exmidontri  7195  exmidontri2or  7199