Theorem regexmid 4519

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 4521. (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 2170	. . 3 |- { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) }
2	1	regexmidlemm 4516	. 2 |- E. y y e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) }
3		pp0ex 4175	. . . 4 |- { (/) , { (/) } } e. _V
4	3	rabex 4133	. . 3 |- { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } e. _V
5		eleq2 2234	. . . . 5 |- ( x = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } -> ( y e. x <-> y e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } ) )
6	5	exbidv 1818	. . . 4 |- ( x = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } -> ( E. y y e. x <-> E. y y e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } ) )
7		eleq2 2234	. . . . . . . . 9 |- ( x = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } -> ( z e. x <-> z e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } ) )
8	7	notbid 662	. . . . . . . 8 |- ( x = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } -> ( -. z e. x <-> -. z e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } ) )
9	8	imbi2d 229	. . . . . . 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 1817	. . . . . 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 470	. . . . 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 1818	. . . 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 233	. . 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 2784	. 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 4517	. 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 103   \/ wo 703   A.wal 1346   = wceq 1348   E.wex 1485   e. wcel 2141   {crab 2452   (/)c0 3414   {csn 3583   {cpr 3584

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

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  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

This theorem is referenced by: (None)