Theorem exmid1dc 4131

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 4125 or ordtriexmid 4445. 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 822	. . . . . . 7 |- (DECID x = { (/) } -> ( x = { (/) } \/ -. x = { (/) } ) )
3	1, 2	syl 14	. . . . . 6 |- ( ( ph /\ x C_ { (/) } ) -> ( x = { (/) } \/ -. x = { (/) } ) )
4		df-ne 2310	. . . . . . . . 9 |- ( x =/= { (/) } <-> -. x = { (/) } )
5		pwntru 4130	. . . . . . . . . 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 1847	. 2 |- ( ph -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
14		exmid01 4129	. 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 820   A.wal 1330   = wceq 1332   =/= wne 2309   C_ wss 3076   (/)c0 3368   {csn 3532  EXMIDwem 4126

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-nul 4062

This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089  df-nul 3369  df-sn 3538  df-exmid 4127

This theorem is referenced by:  exmidonfin  7067  exmidaclem  7081