Theorem reg2exmidlema 4638

Description: Lemma for reg2exmid 4640. 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 529	. . . . . . 7 |- ( ( ( u e. A /\ A. v e. A u C_ v ) /\ u = { (/) } ) -> A. v e. A u C_ v )
2		sseq1 3251	. . . . . . . . 9 |- ( u = { (/) } -> ( u C_ v <-> { (/) } C_ v ) )
3	2	ralbidv 2533	. . . . . . . 8 |- ( u = { (/) } -> ( A. v e. A u C_ v <-> A. v e. A { (/) } C_ v ) )
4	3	adantl 277	. . . . . . 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 147	. . . . . 6 |- ( ( ( u e. A /\ A. v e. A u C_ v ) /\ u = { (/) } ) -> A. v e. A { (/) } C_ v )
6		0ex 4221	. . . . . . . 8 |- (/) e. _V
7	6	snss 3813	. . . . . . 7 |- ( (/) e. v <-> { (/) } C_ v )
8	7	ralbii 2539	. . . . . 6 |- ( A. v e. A (/) e. v <-> A. v e. A { (/) } C_ v )
9	5, 8	sylibr 134	. . . . 5 |- ( ( ( u e. A /\ A. v e. A u C_ v ) /\ u = { (/) } ) -> A. v e. A (/) e. v )
10		noel 3500	. . . . . 6 |- -. (/) e. (/)
11		eqid 2231	. . . . . . . . . . . 12 |- (/) = (/)
12	11	jctl 314	. . . . . . . . . . 11 |- ( ph -> ( (/) = (/) /\ ph ) )
13	12	olcd 742	. . . . . . . . . 10 |- ( ph -> ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) )
14	6	prid1 3781	. . . . . . . . . 10 |- (/) e. { (/) , { (/) } }
15	13, 14	jctil 312	. . . . . . . . 9 |- ( ph -> ( (/) e. { (/) , { (/) } } /\ ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) ) )
16		eqeq1 2238	. . . . . . . . . . 11 |- ( x = (/) -> ( x = { (/) } <-> (/) = { (/) } ) )
17		eqeq1 2238	. . . . . . . . . . . 12 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
18	17	anbi1d 465	. . . . . . . . . . 11 |- ( x = (/) -> ( ( x = (/) /\ ph ) <-> ( (/) = (/) /\ ph ) ) )
19	16, 18	orbi12d 801	. . . . . . . . . 10 |- ( x = (/) -> ( ( x = { (/) } \/ ( x = (/) /\ ph ) ) <-> ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) ) )
20		regexmidlemm.a	. . . . . . . . . 10 |- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }
21	19, 20	elrab2 2966	. . . . . . . . 9 |- ( (/) e. A <-> ( (/) e. { (/) , { (/) } } /\ ( (/) = { (/) } \/ ( (/) = (/) /\ ph ) ) ) )
22	15, 21	sylibr 134	. . . . . . . 8 |- ( ph -> (/) e. A )
23		eleq2 2295	. . . . . . . . 9 |- ( v = (/) -> ( (/) e. v <-> (/) e. (/) ) )
24	23	rspcv 2907	. . . . . . . 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 670	. . . . 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 742	. . 3 |- ( ( ( u e. A /\ A. v e. A u C_ v ) /\ u = { (/) } ) -> ( ph \/ -. ph ) )
30		simprr 533	. . . 4 |- ( ( ( u e. A /\ A. v e. A u C_ v ) /\ ( u = (/) /\ ph ) ) -> ph )
31	30	orcd 741	. . 3 |- ( ( ( u e. A /\ A. v e. A u C_ v ) /\ ( u = (/) /\ ph ) ) -> ( ph \/ -. ph ) )
32		eqeq1 2238	. . . . . . 7 |- ( x = u -> ( x = { (/) } <-> u = { (/) } ) )
33		eqeq1 2238	. . . . . . . 8 |- ( x = u -> ( x = (/) <-> u = (/) ) )
34	33	anbi1d 465	. . . . . . 7 |- ( x = u -> ( ( x = (/) /\ ph ) <-> ( u = (/) /\ ph ) ) )
35	32, 34	orbi12d 801	. . . . . 6 |- ( x = u -> ( ( x = { (/) } \/ ( x = (/) /\ ph ) ) <-> ( u = { (/) } \/ ( u = (/) /\ ph ) ) ) )
36	35, 20	elrab2 2966	. . . . 5 |- ( u e. A <-> ( u e. { (/) , { (/) } } /\ ( u = { (/) } \/ ( u = (/) /\ ph ) ) ) )
37	36	simprbi 275	. . . 4 |- ( u e. A -> ( u = { (/) } \/ ( u = (/) /\ ph ) ) )
38	37	adantr 276	. . 3 |- ( ( u e. A /\ A. v e. A u C_ v ) -> ( u = { (/) } \/ ( u = (/) /\ ph ) ) )
39	29, 31, 38	mpjaodan 806	. 2 |- ( ( u e. A /\ A. v e. A u C_ v ) -> ( ph \/ -. ph ) )
40	39	rexlimiva 2646	1 |- ( E. u e. A A. v e. A u C_ v -> ( ph \/ -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   {crab 2515   C_ wss 3201   (/)c0 3496   {csn 3673   {cpr 3674

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-nul 4220

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-sn 3679  df-pr 3680

This theorem is referenced by:  reg2exmid  4640  reg3exmid  4684