Theorem regexmid 4278

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 4280. (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 2081	. . 3 |- { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) }
2	1	regexmidlemm 4275	. 2 |- E. y y e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) }
3		pp0ex 3960	. . . 4 |- { (/) , { (/) } } e. _V
4	3	rabex 3922	. . 3 |- { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } e. _V
5		eleq2 2142	. . . . 5 |- ( x = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } -> ( y e. x <-> y e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } ) )
6	5	exbidv 1746	. . . 4 |- ( x = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } -> ( E. y y e. x <-> E. y y e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } ) )
7		eleq2 2142	. . . . . . . . 9 |- ( x = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } -> ( z e. x <-> z e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } ) )
8	7	notbid 624	. . . . . . . 8 |- ( x = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } -> ( -. z e. x <-> -. z e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } ) )
9	8	imbi2d 228	. . . . . . 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 1745	. . . . . 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 456	. . . . 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 1746	. . . 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 232	. . 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 2653	. 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 4276	. 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 102   \/ wo 661   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   {crab 2352   (/)c0 3251   {csn 3398   {cpr 3399

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405

This theorem is referenced by: (None)