Theorem regexmid 4626

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 4628. (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 2229	. . 3 |- { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) }
2	1	regexmidlemm 4623	. 2 |- E. y y e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) }
3		pp0ex 4272	. . . 4 |- { (/) , { (/) } } e. _V
4	3	rabex 4227	. . 3 |- { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } e. _V
5		eleq2 2293	. . . . 5 |- ( x = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } -> ( y e. x <-> y e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } ) )
6	5	exbidv 1871	. . . 4 |- ( x = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } -> ( E. y y e. x <-> E. y y e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } ) )
7		eleq2 2293	. . . . . . . . 9 |- ( x = { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } -> ( z e. x <-> z e. { w e. { (/) , { (/) } } | ( w = { (/) } \/ ( w = (/) /\ ph ) ) } ) )
8	7	notbid 671	. . . . . . . 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 1870	. . . . . 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 1871	. . . 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 2855	. 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 4624	. 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 713   A.wal 1393   = wceq 1395   E.wex 1538   e. wcel 2200   {crab 2512   (/)c0 3491   {csn 3666   {cpr 3667

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673

This theorem is referenced by: (None)