Theorem exmid1dc 4229

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 4222 or ordtriexmid 4553. 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 837	. . . . . . 7 |- (DECID x = { (/) } -> ( x = { (/) } \/ -. x = { (/) } ) )
3	1, 2	syl 14	. . . . . 6 |- ( ( ph /\ x C_ { (/) } ) -> ( x = { (/) } \/ -. x = { (/) } ) )
4		df-ne 2365	. . . . . . . . 9 |- ( x =/= { (/) } <-> -. x = { (/) } )
5		pwntru 4228	. . . . . . . . . 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 789	. . . . . . 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 730	. . . 4 |- ( ( ph /\ x C_ { (/) } ) -> ( x = (/) \/ x = { (/) } ) )
12	11	ex 115	. . 3 |- ( ph -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
13	12	alrimiv 1885	. 2 |- ( ph -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
14		exmid01 4227	. 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 709  DECID wdc 835   A.wal 1362   = wceq 1364   =/= wne 2364   C_ wss 3153   (/)c0 3446   {csn 3618  EXMIDwem 4223

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-nul 4155

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-v 2762  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3447  df-sn 3624  df-exmid 4224

This theorem is referenced by:  pw1fin  6966  exmidonfin  7254  exmidaclem  7268  exmidontri  7299  exmidontri2or  7303