Theorem regexmid 4450

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 4452. (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 2139	. . 3 |- { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) }
2	1	regexmidlemm 4447	. 2 |- E. y y e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) }
3		pp0ex 4113	. . . 4 |- { (/) , { (/) } } e. _V
4	3	rabex 4072	. . 3 |- { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } e. _V
5		eleq2 2203	. . . . 5 |- ( x = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } -> ( y e. x <-> y e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } ) )
6	5	exbidv 1797	. . . 4 |- ( x = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } -> ( E. y y e. x <-> E. y y e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } ) )
7		eleq2 2203	. . . . . . . . 9 |- ( x = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } -> ( z e. x <-> z e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } ) )
8	7	notbid 656	. . . . . . . 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 1796	. . . . . 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 464	. . . . 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 1797	. . . 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 2740	. 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 4448	. 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 697   A.wal 1329   = wceq 1331   E.wex 1468   e. wcel 1480   {crab 2420   (/)c0 3363   {csn 3527   {cpr 3528

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534

This theorem is referenced by: (None)