Theorem exmid1dc 4290

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 4283 or ordtriexmid 4619. 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 843	. . . . . . 7 |- (DECID x = { (/) } -> ( x = { (/) } \/ -. x = { (/) } ) )
3	1, 2	syl 14	. . . . . 6 |- ( ( ph /\ x C_ { (/) } ) -> ( x = { (/) } \/ -. x = { (/) } ) )
4		df-ne 2403	. . . . . . . . 9 |- ( x =/= { (/) } <-> -. x = { (/) } )
5		pwntru 4289	. . . . . . . . . 10 |- ( ( x C_ { (/) } /\ x =/= { (/) } ) -> x = (/) )
6	5	ex 115	. . . . . . . . 9 |- ( x C_ { (/) } -> ( x =/= { (/) } -> x = (/) ) )
7	4, 6	biimtrrid 153	. . . . . . . 8 |- ( x C_ { (/) } -> ( -. x = { (/) } -> x = (/) ) )
8	7	orim2d 795	. . . . . . 7 |- ( x C_ { (/) } -> ( ( x = { (/) } \/ -. x = { (/) } ) -> ( x = { (/) } \/ x = (/) ) ) )
9	8	adantl 277	. . . . . 6 |- ( ( ph /\ x C_ { (/) } ) -> ( ( x = { (/) } \/ -. x = { (/) } ) -> ( x = { (/) } \/ x = (/) ) ) )
10	3, 9	mpd 13	. . . . 5 |- ( ( ph /\ x C_ { (/) } ) -> ( x = { (/) } \/ x = (/) ) )
11	10	orcomd 736	. . . 4 |- ( ( ph /\ x C_ { (/) } ) -> ( x = (/) \/ x = { (/) } ) )
12	11	ex 115	. . 3 |- ( ph -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
13	12	alrimiv 1922	. 2 |- ( ph -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
14		exmid01 4288	. 2 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
15	13, 14	sylibr 134	1 |- ( ph -> EXMID )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 715  DECID wdc 841   A.wal 1395   = wceq 1397   =/= wne 2402   C_ wss 3200   (/)c0 3494   {csn 3669  EXMIDwem 4284

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-nul 4215

This theorem depends on definitions:  df-bi 117  df-dc 842  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-sn 3675  df-exmid 4285

This theorem is referenced by:  pw1fin  7101  exmidssfi  7130  exmidonfin  7404  exmidaclem  7422  exmidontri  7456  exmidontri2or  7460