Theorem reg2exmidlema 4350

Description: Lemma for reg2exmid 4352. If A has a minimal element (expressed by C_), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)

Hypothesis

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

Assertion

Ref	Expression
reg2exmidlema	|- ( E. u e. A A. v e. A u C_ v -> ( ph \/ -. ph ) )

Distinct variable groups:   ph, x   v, A   ph, u, x   v, u

Allowed substitution hints:   ph( v)   A( x, u)

Proof of Theorem reg2exmidlema

Step	Hyp	Ref	Expression
1		simplr 497	. . . . . . 7 |- ( ( ( u e. A /\ A. v e. A u C_ v ) /\ u = { (/) } ) -> A. v e. A u C_ v )
2		sseq1 3047	. . . . . . . . 9 |- ( u = { (/) } -> ( u C_ v <-> { (/) } C_ v ) )
3	2	ralbidv 2380	. . . . . . . 8 |- ( u = { (/) } -> ( A. v e. A u C_ v <-> A. v e. A { (/) } C_ v ) )
4	3	adantl 271	. . . . . . 7 |- ( ( ( u e. A /\ A. v e. A u C_ v ) /\ u = { (/) } ) -> ( A. v e. A u C_ v <-> A. v e. A { (/) } C_ v ) )
5	1, 4	mpbid 145	. . . . . 6 |- ( ( ( u e. A /\ A. v e. A u C_ v ) /\ u = { (/) } ) -> A. v e. A { (/) } C_ v )
6		0ex 3966	. . . . . . . 8 |- (/) e. _V
7	6	snss 3566	. . . . . . 7 |- ( (/) e. v <-> { (/) } C_ v )
8	7	ralbii 2384	. . . . . 6 |- ( A. v e. A (/) e. v <-> A. v e. A { (/) } C_ v )
9	5, 8	sylibr 132	. . . . 5 |- ( ( ( u e. A /\ A. v e. A u C_ v ) /\ u = { (/) } ) -> A. v e. A (/) e. v )
10		noel 3290	. . . . . 6 |- -. (/) e. (/)
11		eqid 2088	. . . . . . . . . . . 12 |- (/) = (/)
12	11	jctl 307	. . . . . . . . . . 11 |- ( ph -> ( (/) = (/) /\ ph ) )
13	12	olcd 688	. . . . . . . . . 10 |- ( ph -> ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) )
14	6	prid1 3548	. . . . . . . . . 10 |- (/) e. { (/) , { (/) } }
15	13, 14	jctil 305	. . . . . . . . 9 |- ( ph -> ( (/) e. { (/) , { (/) } } /\ ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) ) )
16		eqeq1 2094	. . . . . . . . . . 11 |- ( x = (/) -> ( x = { (/) } <-> (/) = { (/) } ) )
17		eqeq1 2094	. . . . . . . . . . . 12 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
18	17	anbi1d 453	. . . . . . . . . . 11 |- ( x = (/) -> ( ( x = (/) /\ ph ) <-> ( (/) = (/) /\ ph ) ) )
19	16, 18	orbi12d 742	. . . . . . . . . 10 |- ( x = (/) -> ( ( x = { (/) } \/ ( x = (/) /\ ph ) ) <-> ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) ) )
20		regexmidlemm.a	. . . . . . . . . 10 |- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }
21	19, 20	elrab2 2774	. . . . . . . . 9 |- ( (/) e. A <-> ( (/) e. { (/) , { (/) } } /\ ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) ) )
22	15, 21	sylibr 132	. . . . . . . 8 |- ( ph -> (/) e. A )
23		eleq2 2151	. . . . . . . . 9 |- ( v = (/) -> ( (/) e. v <-> (/) e. (/) ) )
24	23	rspcv 2718	. . . . . . . 8 |- ( (/) e. A -> ( A. v e. A (/) e. v -> (/) e. (/) ) )
25	22, 24	syl 14	. . . . . . 7 |- ( ph -> ( A. v e. A (/) e. v -> (/) e. (/) ) )
26	25	com12 30	. . . . . 6 |- ( A. v e. A (/) e. v -> ( ph -> (/) e. (/) ) )
27	10, 26	mtoi 625	. . . . 5 |- ( A. v e. A (/) e. v -> -. ph )
28	9, 27	syl 14	. . . 4 |- ( ( ( u e. A /\ A. v e. A u C_ v ) /\ u = { (/) } ) -> -. ph )
29	28	olcd 688	. . 3 |- ( ( ( u e. A /\ A. v e. A u C_ v ) /\ u = { (/) } ) -> ( ph \/ -. ph ) )
30		simprr 499	. . . 4 |- ( ( ( u e. A /\ A. v e. A u C_ v ) /\ ( u = (/) /\ ph ) ) -> ph )
31	30	orcd 687	. . 3 |- ( ( ( u e. A /\ A. v e. A u C_ v ) /\ ( u = (/) /\ ph ) ) -> ( ph \/ -. ph ) )
32		eqeq1 2094	. . . . . . 7 |- ( x = u -> ( x = { (/) } <-> u = { (/) } ) )
33		eqeq1 2094	. . . . . . . 8 |- ( x = u -> ( x = (/) <-> u = (/) ) )
34	33	anbi1d 453	. . . . . . 7 |- ( x = u -> ( ( x = (/) /\ ph ) <-> ( u = (/) /\ ph ) ) )
35	32, 34	orbi12d 742	. . . . . 6 |- ( x = u -> ( ( x = { (/) } \/ ( x = (/) /\ ph ) ) <-> ( u = { (/) } \/ ( u = (/) /\ ph ) ) ) )
36	35, 20	elrab2 2774	. . . . 5 |- ( u e. A <-> ( u e. { (/) , { (/) } } /\ ( u = { (/) } \/ ( u = (/) /\ ph ) ) ) )
37	36	simprbi 269	. . . 4 |- ( u e. A -> ( u = { (/) } \/ ( u = (/) /\ ph ) ) )
38	37	adantr 270	. . 3 |- ( ( u e. A /\ A. v e. A u C_ v ) -> ( u = { (/) } \/ ( u = (/) /\ ph ) ) )
39	29, 31, 38	mpjaodan 747	. 2 |- ( ( u e. A /\ A. v e. A u C_ v ) -> ( ph \/ -. ph ) )
40	39	rexlimiva 2484	1 |- ( E. u e. A A. v e. A u C_ v -> ( ph \/ -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 664   = wceq 1289   e. wcel 1438   A.wral 2359   E.wrex 2360   {crab 2363   C_ wss 2999   (/)c0 3286   {csn 3446   {cpr 3447

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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3965

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-sn 3452  df-pr 3453

This theorem is referenced by:  reg2exmid  4352  reg3exmid  4395