Theorem exmid1dc 4332

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 4325 or ordtriexmid 4663. 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 848	. . . . . . 7 |- (DECID x = { (/) } -> ( x = { (/) } \/ -. x = { (/) } ) )
3	1, 2	syl 14	. . . . . 6 |- ( ( ph /\ x C_ { (/) } ) -> ( x = { (/) } \/ -. x = { (/) } ) )
4		df-ne 2421	. . . . . . . . 9 |- ( x =/= { (/) } <-> -. x = { (/) } )
5		pwntru 4331	. . . . . . . . . 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 800	. . . . . . 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 741	. . . 4 |- ( ( ph /\ x C_ { (/) } ) -> ( x = (/) \/ x = { (/) } ) )
12	11	ex 115	. . 3 |- ( ph -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
13	12	alrimiv 1927	. 2 |- ( ph -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
14		exmid01 4330	. 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 720  DECID wdc 846   A.wal 1400   = wceq 1402   =/= wne 2420   C_ wss 3220   (/)c0 3520   {csn 3705  EXMIDwem 4326

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-nul 4254

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3711  df-exmid 4327

This theorem is referenced by:  pw1fin  7207  exmidssfi  7236  exmidonfin  7536  exmidaclem  7554  exmidontri  7588  exmidontri2or  7592