Theorem reg3exmid 4564

Description: If any inhabited set satisfying df-wetr 4319 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 2170	. . 3 |- { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) }
2	1	regexmidlemm 4516	. 2 |- E. w w e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) }
3	1	reg3exmidlemwe 4563	. . 3 |- _E We { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) }
4		pp0ex 4175	. . . . 5 |- { (/) , { (/) } } e. _V
5	4	rabex 4133	. . . 4 |- { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } e. _V
6		weeq2 4342	. . . . . 6 |- ( z = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } -> ( _E We z <-> _E We { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } ) )
7		eleq2 2234	. . . . . . 7 |- ( z = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } -> ( w e. z <-> w e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } ) )
8	7	exbidv 1818	. . . . . 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 470	. . . . 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 2665	. . . . . 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 2674	. . . . 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 233	. . . 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 2784	. . 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 422	. 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 4518	. 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 103   \/ wo 703   = wceq 1348   E.wex 1485   e. wcel 2141   A.wral 2448   E.wrex 2449   {crab 2452   C_ wss 3121   (/)c0 3414   {csn 3583   {cpr 3584   _E cep 4272   We wwe 4315

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-eprel 4274  df-frfor 4316  df-frind 4317  df-wetr 4319

This theorem is referenced by: (None)