Theorem reg3exmidlemwe 4321

Description: Lemma for reg3exmid 4322. Our counterexample A satisfies We. (Contributed by Jim Kingdon, 3-Oct-2021.)

Hypothesis

Ref	Expression
reg3exmidlemwe.a	|- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }

Assertion

Ref	Expression
reg3exmidlemwe	|- _E We A

Distinct variable group:   ph, x

Allowed substitution hint:   A( x)

Proof of Theorem reg3exmidlemwe

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zfregfr 4316	. 2 |- _E Fr A
2		epel 4047	. . . . . 6 |- ( a _E b <-> a e. b )
3		epel 4047	. . . . . 6 |- ( b _E c <-> b e. c )
4	2, 3	anbi12i 447	. . . . 5 |- ( ( a _E b /\ b _E c ) <-> ( a e. b /\ b e. c ) )
5		simpr 108	. . . . . 6 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> ( a e. b /\ b e. c ) )
6		elirr 4284	. . . . . . . 8 |- -. { (/) } e. { (/) }
7		simprr 498	. . . . . . . . . 10 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> b e. c )
8		noel 3255	. . . . . . . . . . . . 13 |- -. a e. (/)
9		eleq2 2142	. . . . . . . . . . . . 13 |- ( b = (/) -> ( a e. b <-> a e. (/) ) )
10	8, 9	mtbiri 632	. . . . . . . . . . . 12 |- ( b = (/) -> -. a e. b )
11		simprl 497	. . . . . . . . . . . 12 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> a e. b )
12	10, 11	nsyl3 588	. . . . . . . . . . 11 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> -. b = (/) )
13		elrabi 2746	. . . . . . . . . . . . . . . 16 |- ( b e. { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) } -> b e. { (/) , { (/) } } )
14		reg3exmidlemwe.a	. . . . . . . . . . . . . . . 16 |- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }
15	13, 14	eleq2s 2173	. . . . . . . . . . . . . . 15 |- ( b e. A -> b e. { (/) , { (/) } } )
16		elpri 3421	. . . . . . . . . . . . . . 15 |- ( b e. { (/) , { (/) } } -> ( b = (/) \/ b = { (/) } ) )
17	15, 16	syl 14	. . . . . . . . . . . . . 14 |- ( b e. A -> ( b = (/) \/ b = { (/) } ) )
18	17	orcomd 680	. . . . . . . . . . . . 13 |- ( b e. A -> ( b = { (/) } \/ b = (/) ) )
19	18	3ad2ant2 960	. . . . . . . . . . . 12 |- ( ( a e. A /\ b e. A /\ c e. A ) -> ( b = { (/) } \/ b = (/) ) )
20	19	adantr 270	. . . . . . . . . . 11 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> ( b = { (/) } \/ b = (/) ) )
21	12, 20	ecased 1280	. . . . . . . . . 10 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> b = { (/) } )
22		noel 3255	. . . . . . . . . . . . 13 |- -. b e. (/)
23		eleq2 2142	. . . . . . . . . . . . 13 |- ( c = (/) -> ( b e. c <-> b e. (/) ) )
24	22, 23	mtbiri 632	. . . . . . . . . . . 12 |- ( c = (/) -> -. b e. c )
25	24, 7	nsyl3 588	. . . . . . . . . . 11 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> -. c = (/) )
26		elrabi 2746	. . . . . . . . . . . . . . . 16 |- ( c e. { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) } -> c e. { (/) , { (/) } } )
27	26, 14	eleq2s 2173	. . . . . . . . . . . . . . 15 |- ( c e. A -> c e. { (/) , { (/) } } )
28		vex 2604	. . . . . . . . . . . . . . . 16 |- c e. _V
29	28	elpr 3419	. . . . . . . . . . . . . . 15 |- ( c e. { (/) , { (/) } } <-> ( c = (/) \/ c = { (/) } ) )
30	27, 29	sylib 120	. . . . . . . . . . . . . 14 |- ( c e. A -> ( c = (/) \/ c = { (/) } ) )
31	30	orcomd 680	. . . . . . . . . . . . 13 |- ( c e. A -> ( c = { (/) } \/ c = (/) ) )
32	31	3ad2ant3 961	. . . . . . . . . . . 12 |- ( ( a e. A /\ b e. A /\ c e. A ) -> ( c = { (/) } \/ c = (/) ) )
33	32	adantr 270	. . . . . . . . . . 11 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> ( c = { (/) } \/ c = (/) ) )
34	25, 33	ecased 1280	. . . . . . . . . 10 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> c = { (/) } )
35	7, 21, 34	3eltr3d 2161	. . . . . . . . 9 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> { (/) } e. { (/) } )
36	35	ex 113	. . . . . . . 8 |- ( ( a e. A /\ b e. A /\ c e. A ) -> ( ( a e. b /\ b e. c ) -> { (/) } e. { (/) } ) )
37	6, 36	mtoi 622	. . . . . . 7 |- ( ( a e. A /\ b e. A /\ c e. A ) -> -. ( a e. b /\ b e. c ) )
38	37	adantr 270	. . . . . 6 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> -. ( a e. b /\ b e. c ) )
39	5, 38	pm2.21dd 582	. . . . 5 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> a _E c )
40	4, 39	sylan2b 281	. . . 4 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a _E b /\ b _E c ) ) -> a _E c )
41	40	ex 113	. . 3 |- ( ( a e. A /\ b e. A /\ c e. A ) -> ( ( a _E b /\ b _E c ) -> a _E c ) )
42	41	rgen3 2448	. 2 |- A. a e. A A. b e. A A. c e. A ( ( a _E b /\ b _E c ) -> a _E c )
43		df-wetr 4089	. 2 |- ( _E We A <-> ( _E Fr A /\ A. a e. A A. b e. A A. c e. A ( ( a _E b /\ b _E c ) -> a _E c ) ) )
44	1, 42, 43	mpbir2an 883	1 |- _E We A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   \/ wo 661   /\ w3a 919   = wceq 1284   e. wcel 1433   A.wral 2348   {crab 2352   (/)c0 3251   {csn 3398   {cpr 3399   class class class wbr 3785   _E cep 4042   Fr wfr 4083   We wwe 4085

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-eprel 4044  df-frfor 4086  df-frind 4087  df-wetr 4089

This theorem is referenced by:  reg3exmid  4322