Theorem regexmid 4662

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 4664. (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 2234	. . 3 |- { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) }
2	1	regexmidlemm 4659	. 2 |- E. y y e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) }
3		pp0ex 4307	. . . 4 |- { (/) , { (/) } } e. _V
4	3	rabex 4261	. . 3 |- { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } e. _V
5		eleq2 2298	. . . . 5 |- ( x = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } -> ( y e. x <-> y e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } ) )
6	5	exbidv 1874	. . . 4 |- ( x = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } -> ( E. y y e. x <-> E. y y e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } ) )
7		eleq2 2298	. . . . . . . . 9 |- ( x = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } -> ( z e. x <-> z e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } ) )
8	7	notbid 673	. . . . . . . 8 |- ( x = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } -> ( -. z e. x <-> -. z e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } ) )
9	8	imbi2d 230	. . . . . . 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 1873	. . . . . 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 473	. . . . 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 1874	. . . 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 234	. . 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 2871	. 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 4660	. 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 104   \/ wo 716   A.wal 1396   = wceq 1398   E.wex 1541   e. wcel 2205   {crab 2526   (/)c0 3512   {csn 3694   {cpr 3695

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rab 2531  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701

This theorem is referenced by: (None)