Theorem regexmidlem1 4276

Description: Lemma for regexmid 4278. If A has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2019.)

Hypothesis

Ref	Expression
regexmidlemm.a	|- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }

Assertion

Ref	Expression
regexmidlem1	|- ( E. y ( y e. A /\ A. z ( z e. y -> -. z e. A ) ) -> ( ph \/ -. ph ) )

Distinct variable groups:   y, A, z   ph, x, y

Allowed substitution hints:   ph( z)   A( x)

Proof of Theorem regexmidlem1

Step	Hyp	Ref	Expression
1		eqeq1 2087	. . . . . . 7 |- ( x = y -> ( x = { (/) } <-> y = { (/) } ) )
2		eqeq1 2087	. . . . . . . 8 |- ( x = y -> ( x = (/) <-> y = (/) ) )
3	2	anbi1d 452	. . . . . . 7 |- ( x = y -> ( ( x = (/) /\ ph ) <-> ( y = (/) /\ ph ) ) )
4	1, 3	orbi12d 739	. . . . . 6 |- ( x = y -> ( ( x = { (/) } \/ ( x = (/) /\ ph ) ) <-> ( y = { (/) } \/ ( y = (/) /\ ph ) ) ) )
5		regexmidlemm.a	. . . . . 6 |- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }
6	4, 5	elrab2 2751	. . . . 5 |- ( y e. A <-> ( y e. { (/) , { (/) } } /\ ( y = { (/) } \/ ( y = (/) /\ ph ) ) ) )
7	6	simprbi 269	. . . 4 |- ( y e. A -> ( y = { (/) } \/ ( y = (/) /\ ph ) ) )
8		0ex 3905	. . . . . . . . 9 |- (/) e. _V
9	8	snid 3425	. . . . . . . 8 |- (/) e. { (/) }
10		eleq2 2142	. . . . . . . 8 |- ( y = { (/) } -> ( (/) e. y <-> (/) e. { (/) } ) )
11	9, 10	mpbiri 166	. . . . . . 7 |- ( y = { (/) } -> (/) e. y )
12		eleq1 2141	. . . . . . . . 9 |- ( z = (/) -> ( z e. y <-> (/) e. y ) )
13		eleq1 2141	. . . . . . . . . 10 |- ( z = (/) -> ( z e. A <-> (/) e. A ) )
14	13	notbid 624	. . . . . . . . 9 |- ( z = (/) -> ( -. z e. A <-> -. (/) e. A ) )
15	12, 14	imbi12d 232	. . . . . . . 8 |- ( z = (/) -> ( ( z e. y -> -. z e. A ) <-> ( (/) e. y -> -. (/) e. A ) ) )
16	8, 15	spcv 2691	. . . . . . 7 |- ( A. z ( z e. y -> -. z e. A ) -> ( (/) e. y -> -. (/) e. A ) )
17	11, 16	syl5com 29	. . . . . 6 |- ( y = { (/) } -> ( A. z ( z e. y -> -. z e. A ) -> -. (/) e. A ) )
18	8	prid1 3498	. . . . . . . . . 10 |- (/) e. { (/) , { (/) } }
19		eqeq1 2087	. . . . . . . . . . . 12 |- ( x = (/) -> ( x = { (/) } <-> (/) = { (/) } ) )
20		eqeq1 2087	. . . . . . . . . . . . 13 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
21	20	anbi1d 452	. . . . . . . . . . . 12 |- ( x = (/) -> ( ( x = (/) /\ ph ) <-> ( (/) = (/) /\ ph ) ) )
22	19, 21	orbi12d 739	. . . . . . . . . . 11 |- ( x = (/) -> ( ( x = { (/) } \/ ( x = (/) /\ ph ) ) <-> ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) ) )
23	22, 5	elrab2 2751	. . . . . . . . . 10 |- ( (/) e. A <-> ( (/) e. { (/) , { (/) } } /\ ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) ) )
24	18, 23	mpbiran 881	. . . . . . . . 9 |- ( (/) e. A <-> ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) )
25		pm2.46 690	. . . . . . . . 9 |- ( -. ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) -> -. ( (/) = (/) /\ ph ) )
26	24, 25	sylnbi 635	. . . . . . . 8 |- ( -. (/) e. A -> -. ( (/) = (/) /\ ph ) )
27		eqid 2081	. . . . . . . . 9 |- (/) = (/)
28	27	biantrur 297	. . . . . . . 8 |- ( ph <-> ( (/) = (/) /\ ph ) )
29	26, 28	sylnibr 634	. . . . . . 7 |- ( -. (/) e. A -> -. ph )
30	29	olcd 685	. . . . . 6 |- ( -. (/) e. A -> ( ph \/ -. ph ) )
31	17, 30	syl6 33	. . . . 5 |- ( y = { (/) } -> ( A. z ( z e. y -> -. z e. A ) -> ( ph \/ -. ph ) ) )
32		orc 665	. . . . . . 7 |- ( ph -> ( ph \/ -. ph ) )
33	32	adantl 271	. . . . . 6 |- ( ( y = (/) /\ ph ) -> ( ph \/ -. ph ) )
34	33	a1d 22	. . . . 5 |- ( ( y = (/) /\ ph ) -> ( A. z ( z e. y -> -. z e. A ) -> ( ph \/ -. ph ) ) )
35	31, 34	jaoi 668	. . . 4 |- ( ( y = { (/) } \/ ( y = (/) /\ ph ) ) -> ( A. z ( z e. y -> -. z e. A ) -> ( ph \/ -. ph ) ) )
36	7, 35	syl 14	. . 3 |- ( y e. A -> ( A. z ( z e. y -> -. z e. A ) -> ( ph \/ -. ph ) ) )
37	36	imp 122	. 2 |- ( ( y e. A /\ A. z ( z e. y -> -. z e. A ) ) -> ( ph \/ -. ph ) )
38	37	exlimiv 1529	1 |- ( E. y ( y e. A /\ A. z ( z e. y -> -. z e. A ) ) -> ( 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-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-nul 3904

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-nul 3252  df-sn 3404  df-pr 3405

This theorem is referenced by:  regexmid  4278  nnregexmid  4360