Theorem reg3exmid 4671

Description: If any inhabited set satisfying df-wetr 4424 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 2229	. . 3 |- { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) }
2	1	regexmidlemm 4623	. 2 |- E. w w e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) }
3	1	reg3exmidlemwe 4670	. . 3 |- _E We { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) }
4		pp0ex 4272	. . . . 5 |- { (/) , { (/) } } e. _V
5	4	rabex 4227	. . . 4 |- { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } e. _V
6		weeq2 4447	. . . . . 6 |- ( z = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } -> ( _E We z <-> _E We { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } ) )
7		eleq2 2293	. . . . . . 7 |- ( z = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } -> ( w e. z <-> w e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } ) )
8	7	exbidv 1871	. . . . . 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 473	. . . . 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 2728	. . . . . 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 2741	. . . . 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 234	. . . 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 2855	. . 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 424	. 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 4625	. 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 104   \/ wo 713   = wceq 1395   E.wex 1538   e. wcel 2200   A.wral 2508   E.wrex 2509   {crab 2512   C_ wss 3197   (/)c0 3491   {csn 3666   {cpr 3667   _E cep 4377   We wwe 4420

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-setind 4628

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-eprel 4379  df-frfor 4421  df-frind 4422  df-wetr 4424

This theorem is referenced by: (None)