Theorem regexmidlem1 4443

Description: Lemma for regexmid 4445. 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 2144	. . . . . . 7 |- ( x = y -> ( x = { (/) } <-> y = { (/) } ) )
2		eqeq1 2144	. . . . . . . 8 |- ( x = y -> ( x = (/) <-> y = (/) ) )
3	2	anbi1d 460	. . . . . . 7 |- ( x = y -> ( ( x = (/) /\ ph ) <-> ( y = (/) /\ ph ) ) )
4	1, 3	orbi12d 782	. . . . . 6 |- ( x = y -> ( ( x = { (/) } \/ ( x = (/) /\ ph ) ) <-> ( y = { (/) } \/ ( y = (/) /\ ph ) ) ) )
5		regexmidlemm.a	. . . . . 6 |- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }
6	4, 5	elrab2 2838	. . . . 5 |- ( y e. A <-> ( y e. { (/) , { (/) } } /\ ( y = { (/) } \/ ( y = (/) /\ ph ) ) ) )
7	6	simprbi 273	. . . 4 |- ( y e. A -> ( y = { (/) } \/ ( y = (/) /\ ph ) ) )
8		0ex 4050	. . . . . . . . 9 |- (/) e. _V
9	8	snid 3551	. . . . . . . 8 |- (/) e. { (/) }
10		eleq2 2201	. . . . . . . 8 |- ( y = { (/) } -> ( (/) e. y <-> (/) e. { (/) } ) )
11	9, 10	mpbiri 167	. . . . . . 7 |- ( y = { (/) } -> (/) e. y )
12		eleq1 2200	. . . . . . . . 9 |- ( z = (/) -> ( z e. y <-> (/) e. y ) )
13		eleq1 2200	. . . . . . . . . 10 |- ( z = (/) -> ( z e. A <-> (/) e. A ) )
14	13	notbid 656	. . . . . . . . 9 |- ( z = (/) -> ( -. z e. A <-> -. (/) e. A ) )
15	12, 14	imbi12d 233	. . . . . . . 8 |- ( z = (/) -> ( ( z e. y -> -. z e. A ) <-> ( (/) e. y -> -. (/) e. A ) ) )
16	8, 15	spcv 2774	. . . . . . 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 3624	. . . . . . . . . 10 |- (/) e. { (/) , { (/) } }
19		eqeq1 2144	. . . . . . . . . . . 12 |- ( x = (/) -> ( x = { (/) } <-> (/) = { (/) } ) )
20		eqeq1 2144	. . . . . . . . . . . . 13 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
21	20	anbi1d 460	. . . . . . . . . . . 12 |- ( x = (/) -> ( ( x = (/) /\ ph ) <-> ( (/) = (/) /\ ph ) ) )
22	19, 21	orbi12d 782	. . . . . . . . . . 11 |- ( x = (/) -> ( ( x = { (/) } \/ ( x = (/) /\ ph ) ) <-> ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) ) )
23	22, 5	elrab2 2838	. . . . . . . . . 10 |- ( (/) e. A <-> ( (/) e. { (/) , { (/) } } /\ ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) ) )
24	18, 23	mpbiran 924	. . . . . . . . 9 |- ( (/) e. A <-> ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) )
25		pm2.46 728	. . . . . . . . 9 |- ( -. ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) -> -. ( (/) = (/) /\ ph ) )
26	24, 25	sylnbi 667	. . . . . . . 8 |- ( -. (/) e. A -> -. ( (/) = (/) /\ ph ) )
27		eqid 2137	. . . . . . . . 9 |- (/) = (/)
28	27	biantrur 301	. . . . . . . 8 |- ( ph <-> ( (/) = (/) /\ ph ) )
29	26, 28	sylnibr 666	. . . . . . 7 |- ( -. (/) e. A -> -. ph )
30	29	olcd 723	. . . . . 6 |- ( -. (/) e. A -> ( ph \/ -. ph ) )
31	17, 30	syl6 33	. . . . 5 |- ( y = { (/) } -> ( A. z ( z e. y -> -. z e. A ) -> ( ph \/ -. ph ) ) )
32		orc 701	. . . . . . 7 |- ( ph -> ( ph \/ -. ph ) )
33	32	adantl 275	. . . . . 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 705	. . . 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 123	. 2 |- ( ( y e. A /\ A. z ( z e. y -> -. z e. A ) ) -> ( ph \/ -. ph ) )
38	37	exlimiv 1577	1 |- ( E. y ( y e. A /\ A. z ( z e. y -> -. z e. A ) ) -> ( ph \/ -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 697   A.wal 1329   = wceq 1331   E.wex 1468   e. wcel 1480   {crab 2418   (/)c0 3358   {csn 3522   {cpr 3523

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-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-nul 4049

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-nul 3359  df-sn 3528  df-pr 3529

This theorem is referenced by:  regexmid  4445  nnregexmid  4529