Theorem regexmidlem1 4565

Description: Lemma for regexmid 4567. 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 2200	. . . . . . 7 |- ( x = y -> ( x = { (/) } <-> y = { (/) } ) )
2		eqeq1 2200	. . . . . . . 8 |- ( x = y -> ( x = (/) <-> y = (/) ) )
3	2	anbi1d 465	. . . . . . 7 |- ( x = y -> ( ( x = (/) /\ ph ) <-> ( y = (/) /\ ph ) ) )
4	1, 3	orbi12d 794	. . . . . 6 |- ( x = y -> ( ( x = { (/) } \/ ( x = (/) /\ ph ) ) <-> ( y = { (/) } \/ ( y = (/) /\ ph ) ) ) )
5		regexmidlemm.a	. . . . . 6 |- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }
6	4, 5	elrab2 2919	. . . . 5 |- ( y e. A <-> ( y e. { (/) , { (/) } } /\ ( y = { (/) } \/ ( y = (/) /\ ph ) ) ) )
7	6	simprbi 275	. . . 4 |- ( y e. A -> ( y = { (/) } \/ ( y = (/) /\ ph ) ) )
8		0ex 4156	. . . . . . . . 9 |- (/) e. _V
9	8	snid 3649	. . . . . . . 8 |- (/) e. { (/) }
10		eleq2 2257	. . . . . . . 8 |- ( y = { (/) } -> ( (/) e. y <-> (/) e. { (/) } ) )
11	9, 10	mpbiri 168	. . . . . . 7 |- ( y = { (/) } -> (/) e. y )
12		eleq1 2256	. . . . . . . . 9 |- ( z = (/) -> ( z e. y <-> (/) e. y ) )
13		eleq1 2256	. . . . . . . . . 10 |- ( z = (/) -> ( z e. A <-> (/) e. A ) )
14	13	notbid 668	. . . . . . . . 9 |- ( z = (/) -> ( -. z e. A <-> -. (/) e. A ) )
15	12, 14	imbi12d 234	. . . . . . . 8 |- ( z = (/) -> ( ( z e. y -> -. z e. A ) <-> ( (/) e. y -> -. (/) e. A ) ) )
16	8, 15	spcv 2854	. . . . . . 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 3724	. . . . . . . . . 10 |- (/) e. { (/) , { (/) } }
19		eqeq1 2200	. . . . . . . . . . . 12 |- ( x = (/) -> ( x = { (/) } <-> (/) = { (/) } ) )
20		eqeq1 2200	. . . . . . . . . . . . 13 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
21	20	anbi1d 465	. . . . . . . . . . . 12 |- ( x = (/) -> ( ( x = (/) /\ ph ) <-> ( (/) = (/) /\ ph ) ) )
22	19, 21	orbi12d 794	. . . . . . . . . . 11 |- ( x = (/) -> ( ( x = { (/) } \/ ( x = (/) /\ ph ) ) <-> ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) ) )
23	22, 5	elrab2 2919	. . . . . . . . . 10 |- ( (/) e. A <-> ( (/) e. { (/) , { (/) } } /\ ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) ) )
24	18, 23	mpbiran 942	. . . . . . . . 9 |- ( (/) e. A <-> ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) )
25		pm2.46 740	. . . . . . . . 9 |- ( -. ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) -> -. ( (/) = (/) /\ ph ) )
26	24, 25	sylnbi 679	. . . . . . . 8 |- ( -. (/) e. A -> -. ( (/) = (/) /\ ph ) )
27		eqid 2193	. . . . . . . . 9 |- (/) = (/)
28	27	biantrur 303	. . . . . . . 8 |- ( ph <-> ( (/) = (/) /\ ph ) )
29	26, 28	sylnibr 678	. . . . . . 7 |- ( -. (/) e. A -> -. ph )
30	29	olcd 735	. . . . . 6 |- ( -. (/) e. A -> ( ph \/ -. ph ) )
31	17, 30	syl6 33	. . . . 5 |- ( y = { (/) } -> ( A. z ( z e. y -> -. z e. A ) -> ( ph \/ -. ph ) ) )
32		orc 713	. . . . . . 7 |- ( ph -> ( ph \/ -. ph ) )
33	32	adantl 277	. . . . . 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 717	. . . 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 124	. 2 |- ( ( y e. A /\ A. z ( z e. y -> -. z e. A ) ) -> ( ph \/ -. ph ) )
38	37	exlimiv 1609	1 |- ( E. y ( y e. A /\ A. z ( z e. y -> -. z e. A ) ) -> ( ph \/ -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709   A.wal 1362   = wceq 1364   E.wex 1503   e. wcel 2164   {crab 2476   (/)c0 3446   {csn 3618   {cpr 3619

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-nul 4155

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-nul 3447  df-sn 3624  df-pr 3625

This theorem is referenced by:  regexmid  4567  nnregexmid  4653