Theorem regexmid 4219

Description: The axiom of foundation implies excluded middle.

By foundation (or regularity), we mean the principle that every inhabited set has an element which is minimal (when arranged by e.). The statement of foundation here is taken from Metamath Proof Explorer's ax-reg, and is identical (modulo one unnecessary quantifier) to the statement of foundation in Theorem "Foundation implies instances of EM" of [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

For this reason, IZF does not adopt foundation as an axiom and instead replaces it with ax-setind 4220. (Contributed by Jim Kingdon, 3-Sep-2019.)

Hypothesis

Ref	Expression
regexmid.1	|- (E.y y e. x -> E.y(y e. x /\ A.z(z e. y -> -. z e. x)))

Assertion

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

Distinct variable group:   ph,x,y,z

Proof of Theorem regexmid

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2037	. . 3 |- {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) } = {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) }
2	1	regexmidlemm 4217	. 2 |- E.y y e. {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) }
3		pp0ex 3931	. . . 4 |- { (/) , { (/) } } e. _V
4	3	rabex 3892	. . 3 |- {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) } e. _V
5		eleq2 2098	. . . . 5 |- (x = {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) } -> (y e. x <-> y e. {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) }))
6	5	exbidv 1703	. . . 4 |- (x = {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) } -> (E.y y e. x <-> E.y y e. {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) }))
7		eleq2 2098	. . . . . . . . 9 |- (x = {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) } -> (z e. x <-> z e. {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) }))
8	7	notbid 591	. . . . . . . 8 |- (x = {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) } -> (-. z e. x <-> -. z e. {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) }))
9	8	imbi2d 219	. . . . . . 7 |- (x = {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) } -> ((z e. y -> -. z e. x) <-> (z e. y -> -. z e. {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) })))
10	9	albidv 1702	. . . . . 6 |- (x = {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) } -> (A.z(z e. y -> -. z e. x) <-> A.z(z e. y -> -. z e. {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) })))
11	5, 10	anbi12d 442	. . . . 5 |- (x = {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) } -> ((y e. x /\ A.z(z e. y -> -. z e. x)) <-> (y e. {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) } /\ A.z(z e. y -> -. z e. {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) }))))
12	11	exbidv 1703	. . . 4 |- (x = {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) } -> (E.y(y e. x /\ A.z(z e. y -> -. z e. x)) <-> E.y(y e. {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) } /\ A.z(z e. y -> -. z e. {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) }))))
13	6, 12	imbi12d 223	. . 3 |- (x = {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) } -> ((E.y y e. x -> E.y(y e. x /\ A.z(z e. y -> -. z e. x))) <-> (E.y y e. {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) } -> E.y(y e. {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) } /\ A.z(z e. y -> -. z e. {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) })))))
14		regexmid.1	. . 3 |- (E.y y e. x -> E.y(y e. x /\ A.z(z e. y -> -. z e. x)))
15	4, 13, 14	vtocl 2602	. 2 |- (E.y y e. {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) } -> E.y(y e. {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) } /\ A.z(z e. y -> -. z e. {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) })))
16	1	regexmidlem1 4218	. 2 |- (E.y(y e. {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) } /\ A.z(z e. y -> -. z e. {w e. { (/) , { (/) } } | (w = { (/) } \/ (w = (/) /\ ph)) })) -> (ph \/ -. ph))
17	2, 15, 16	mp2b 8	1 |- (ph \/ -. ph)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97   \/ wo 628  A.wal 1240   = wceq 1242  E.wex 1378   e. wcel 1390   {crab 2304   (/)c0 3218   {csn 3367   {cpr 3368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374

This theorem is referenced by: (None)