Theorem reg2exmid 4513

Description: If any inhabited set has a minimal element (when expressed by C_), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)

Hypothesis

Ref	Expression
reg2exmid.1	|- A. z ( E. w w e. z -> E. x e. z A. y e. z x C_ y )

Assertion

Ref	Expression
reg2exmid	|- ( ph \/ -. ph )

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

Proof of Theorem reg2exmid

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2165	. . . 4 |- { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) }
2	1	regexmidlemm 4509	. . 3 |- E. w w e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) }
3		reg2exmid.1	. . . 4 |- A. z ( E. w w e. z -> E. x e. z A. y e. z x C_ y )
4		pp0ex 4168	. . . . . 6 |- { (/) , { (/) } } e. _V
5	4	rabex 4126	. . . . 5 |- { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } e. _V
6		eleq2 2230	. . . . . . 7 |- ( z = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } -> ( w e. z <-> w e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } ) )
7	6	exbidv 1813	. . . . . 6 |- ( z = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } -> ( E. w w e. z <-> E. w w e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } ) )
8		raleq 2661	. . . . . . 7 |- ( z = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } -> ( A. y e. z x C_ y <-> A. y e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } x C_ y ) )
9	8	rexeqbi1dv 2670	. . . . . 6 |- ( 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 ) )
10	7, 9	imbi12d 233	. . . . 5 |- ( z = { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } -> ( ( E. w w e. z -> E. x e. z A. y e. z x C_ y ) <-> ( 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 ) ) )
11	5, 10	spcv 2820	. . . 4 |- ( A. z ( E. w w e. z -> E. x e. z A. y e. z x C_ y ) -> ( 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 ) )
12	3, 11	ax-mp 5	. . 3 |- ( 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	2, 12	ax-mp 5	. 2 |- E. x e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } A. y e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } x C_ y
14	1	reg2exmidlema 4511	. 2 |- ( E. x e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } A. y e. { u e. { (/) , { (/) } } | ( u = { (/) } \/ ( u = (/) /\ ph ) ) } x C_ y -> ( ph \/ -. ph ) )
15	13, 14	ax-mp 5	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698   A.wal 1341   = wceq 1343   E.wex 1480   e. wcel 2136   A.wral 2444   E.wrex 2445   {crab 2448   C_ wss 3116   (/)c0 3409   {csn 3576   {cpr 3577

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583

This theorem is referenced by: (None)