Theorem regexmidlem1 4534

Description: Lemma for regexmid 4536. 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 2184	. . . . . . 7 |- ( x = y -> ( x = { (/) } <-> y = { (/) } ) )
2		eqeq1 2184	. . . . . . . 8 |- ( x = y -> ( x = (/) <-> y = (/) ) )
3	2	anbi1d 465	. . . . . . 7 |- ( x = y -> ( ( x = (/) /\ ph ) <-> ( y = (/) /\ ph ) ) )
4	1, 3	orbi12d 793	. . . . . 6 |- ( x = y -> ( ( x = { (/) } \/ ( x = (/) /\ ph ) ) <-> ( y = { (/) } \/ ( y = (/) /\ ph ) ) ) )
5		regexmidlemm.a	. . . . . 6 |- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }
6	4, 5	elrab2 2898	. . . . 5 |- ( y e. A <-> ( y e. { (/) , { (/) } } /\ ( y = { (/) } \/ ( y = (/) /\ ph ) ) ) )
7	6	simprbi 275	. . . 4 |- ( y e. A -> ( y = { (/) } \/ ( y = (/) /\ ph ) ) )
8		0ex 4132	. . . . . . . . 9 |- (/) e. _V
9	8	snid 3625	. . . . . . . 8 |- (/) e. { (/) }
10		eleq2 2241	. . . . . . . 8 |- ( y = { (/) } -> ( (/) e. y <-> (/) e. { (/) } ) )
11	9, 10	mpbiri 168	. . . . . . 7 |- ( y = { (/) } -> (/) e. y )
12		eleq1 2240	. . . . . . . . 9 |- ( z = (/) -> ( z e. y <-> (/) e. y ) )
13		eleq1 2240	. . . . . . . . . 10 |- ( z = (/) -> ( z e. A <-> (/) e. A ) )
14	13	notbid 667	. . . . . . . . 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 2833	. . . . . . 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 3700	. . . . . . . . . 10 |- (/) e. { (/) , { (/) } }
19		eqeq1 2184	. . . . . . . . . . . 12 |- ( x = (/) -> ( x = { (/) } <-> (/) = { (/) } ) )
20		eqeq1 2184	. . . . . . . . . . . . 13 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
21	20	anbi1d 465	. . . . . . . . . . . 12 |- ( x = (/) -> ( ( x = (/) /\ ph ) <-> ( (/) = (/) /\ ph ) ) )
22	19, 21	orbi12d 793	. . . . . . . . . . 11 |- ( x = (/) -> ( ( x = { (/) } \/ ( x = (/) /\ ph ) ) <-> ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) ) )
23	22, 5	elrab2 2898	. . . . . . . . . 10 |- ( (/) e. A <-> ( (/) e. { (/) , { (/) } } /\ ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) ) )
24	18, 23	mpbiran 940	. . . . . . . . 9 |- ( (/) e. A <-> ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) )
25		pm2.46 739	. . . . . . . . 9 |- ( -. ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) -> -. ( (/) = (/) /\ ph ) )
26	24, 25	sylnbi 678	. . . . . . . 8 |- ( -. (/) e. A -> -. ( (/) = (/) /\ ph ) )
27		eqid 2177	. . . . . . . . 9 |- (/) = (/)
28	27	biantrur 303	. . . . . . . 8 |- ( ph <-> ( (/) = (/) /\ ph ) )
29	26, 28	sylnibr 677	. . . . . . 7 |- ( -. (/) e. A -> -. ph )
30	29	olcd 734	. . . . . 6 |- ( -. (/) e. A -> ( ph \/ -. ph ) )
31	17, 30	syl6 33	. . . . 5 |- ( y = { (/) } -> ( A. z ( z e. y -> -. z e. A ) -> ( ph \/ -. ph ) ) )
32		orc 712	. . . . . . 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 716	. . . 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 1598	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 708   A.wal 1351   = wceq 1353   E.wex 1492   e. wcel 2148   {crab 2459   (/)c0 3424   {csn 3594   {cpr 3595

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-nul 4131

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-nul 3425  df-sn 3600  df-pr 3601

This theorem is referenced by:  regexmid  4536  nnregexmid  4622