Theorem reg3exmid 4322

Description: If any inhabited set satisfying df-wetr 4089 for _E has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.)

Hypothesis

Ref	Expression
reg3exmid.1	|- ( ( _E We z /\ E. w w e. z ) -> E. x e. z A. y e. z x C_ y )

Assertion

Ref	Expression
reg3exmid	|- ( ph \/ -. ph )

Distinct variable groups:   ph, w, z   ph, x, y, z

Proof of Theorem reg3exmid

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2081	. . 3 |- { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) }
2	1	regexmidlemm 4275	. 2 |- E. w w e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) }
3	1	reg3exmidlemwe 4321	. . 3 |- _E We { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) }
4		pp0ex 3960	. . . . 5 |- { (/) , { (/) } } e. _V
5	4	rabex 3922	. . . 4 |- { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } e. _V
6		weeq2 4112	. . . . . 6 |- ( z = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } -> ( _E We z <-> _E We { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } ) )
7		eleq2 2142	. . . . . . 7 |- ( z = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } -> ( w e. z <-> w e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } ) )
8	7	exbidv 1746	. . . . . 6 |- ( z = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } -> ( E. w w e. z <-> E. w w e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } ) )
9	6, 8	anbi12d 456	. . . . 5 |- ( z = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } -> ( ( _E We z /\ E. w w e. z ) <-> ( _E We { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } /\ E. w w e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } ) ) )
10		raleq 2549	. . . . . 6 |- ( z = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } -> ( A. y e. z x C_ y <-> A. y e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } x C_ y ) )
11	10	rexeqbi1dv 2558	. . . . 5 |- ( z = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } -> ( E. x e. z A. y e. z x C_ y <-> E. x e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } A. y e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } x C_ y ) )
12	9, 11	imbi12d 232	. . . 4 |- ( z = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } -> ( ( ( _E We z /\ E. w w e. z ) -> E. x e. z A. y e. z x C_ y ) <-> ( ( _E We { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } /\ E. w w e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } ) -> E. x e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } A. y e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } x C_ y ) ) )
13		reg3exmid.1	. . . 4 |- ( ( _E We z /\ E. w w e. z ) -> E. x e. z A. y e. z x C_ y )
14	5, 12, 13	vtocl 2653	. . 3 |- ( ( _E We { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } /\ E. w w e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } ) -> E. x e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } A. y e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } x C_ y )
15	3, 14	mpan 414	. 2 |- ( E. w w e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } -> E. x e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } A. y e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } x C_ y )
16	1	reg2exmidlema 4277	. 2 |- ( E. x e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } A. y e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } x C_ y -> ( ph \/ -. ph ) )
17	2, 15, 16	mp2b 8	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   \/ wo 661   = wceq 1284   E.wex 1421   e. wcel 1433   A.wral 2348   E.wrex 2349   {crab 2352   C_ wss 2973   (/)c0 3251   {csn 3398   {cpr 3399   _E cep 4042   We wwe 4085

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-eprel 4044  df-frfor 4086  df-frind 4087  df-wetr 4089

This theorem is referenced by: (None)