Theorem reg3exmid 4722

Description: If any inhabited set satisfying df-wetr 4474 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 2238	. . 3 |- { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) }
2	1	regexmidlemm 4674	. 2 |- E. w w e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) }
3	1	reg3exmidlemwe 4721	. . 3 |- _E We { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) }
4		pp0ex 4321	. . . . 5 |- { (/) , { (/) } } e. _V
5	4	rabex 4275	. . . 4 |- { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } e. _V
6		weeq2 4497	. . . . . 6 |- ( z = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } -> ( _E We z <-> _E We { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } ) )
7		eleq2 2302	. . . . . . 7 |- ( z = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } -> ( w e. z <-> w e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } ) )
8	7	exbidv 1878	. . . . . 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 477	. . . . 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 2749	. . . . . 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 2762	. . . . 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 2877	. . 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 428	. 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 4676	. 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 720   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   C_ wss 3220   (/)c0 3520   {csn 3705   {cpr 3706   _E cep 4427   We wwe 4470

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-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-eprel 4429  df-frfor 4471  df-frind 4472  df-wetr 4474

This theorem is referenced by: (None)