Theorem reg3exmidlemwe 4493

Description: Lemma for reg3exmid 4494. 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 4488	. 2 |- _E Fr A
2		epel 4214	. . . . . 6 |- ( a _E b <-> a e. b )
3		epel 4214	. . . . . 6 |- ( b _E c <-> b e. c )
4	2, 3	anbi12i 455	. . . . 5 |- ( ( a _E b /\ b _E c ) <-> ( a e. b /\ b e. c ) )
5		simpr 109	. . . . . 6 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> ( a e. b /\ b e. c ) )
6		elirr 4456	. . . . . . . 8 |- -. { (/) } e. { (/) }
7		simprr 521	. . . . . . . . . 10 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> b e. c )
8		noel 3367	. . . . . . . . . . . . 13 |- -. a e. (/)
9		eleq2 2203	. . . . . . . . . . . . 13 |- ( b = (/) -> ( a e. b <-> a e. (/) ) )
10	8, 9	mtbiri 664	. . . . . . . . . . . 12 |- ( b = (/) -> -. a e. b )
11		simprl 520	. . . . . . . . . . . 12 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> a e. b )
12	10, 11	nsyl3 615	. . . . . . . . . . 11 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> -. b = (/) )
13		elrabi 2837	. . . . . . . . . . . . . . . 16 |- ( b e. { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) } -> b e. { (/) , { (/) } } )
14		reg3exmidlemwe.a	. . . . . . . . . . . . . . . 16 |- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }
15	13, 14	eleq2s 2234	. . . . . . . . . . . . . . 15 |- ( b e. A -> b e. { (/) , { (/) } } )
16		elpri 3550	. . . . . . . . . . . . . . 15 |- ( b e. { (/) , { (/) } } -> ( b = (/) \/ b = { (/) } ) )
17	15, 16	syl 14	. . . . . . . . . . . . . 14 |- ( b e. A -> ( b = (/) \/ b = { (/) } ) )
18	17	orcomd 718	. . . . . . . . . . . . 13 |- ( b e. A -> ( b = { (/) } \/ b = (/) ) )
19	18	3ad2ant2 1003	. . . . . . . . . . . 12 |- ( ( a e. A /\ b e. A /\ c e. A ) -> ( b = { (/) } \/ b = (/) ) )
20	19	adantr 274	. . . . . . . . . . 11 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> ( b = { (/) } \/ b = (/) ) )
21	12, 20	ecased 1327	. . . . . . . . . 10 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> b = { (/) } )
22		noel 3367	. . . . . . . . . . . . 13 |- -. b e. (/)
23		eleq2 2203	. . . . . . . . . . . . 13 |- ( c = (/) -> ( b e. c <-> b e. (/) ) )
24	22, 23	mtbiri 664	. . . . . . . . . . . 12 |- ( c = (/) -> -. b e. c )
25	24, 7	nsyl3 615	. . . . . . . . . . 11 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> -. c = (/) )
26		elrabi 2837	. . . . . . . . . . . . . . . 16 |- ( c e. { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) } -> c e. { (/) , { (/) } } )
27	26, 14	eleq2s 2234	. . . . . . . . . . . . . . 15 |- ( c e. A -> c e. { (/) , { (/) } } )
28		vex 2689	. . . . . . . . . . . . . . . 16 |- c e. _V
29	28	elpr 3548	. . . . . . . . . . . . . . 15 |- ( c e. { (/) , { (/) } } <-> ( c = (/) \/ c = { (/) } ) )
30	27, 29	sylib 121	. . . . . . . . . . . . . 14 |- ( c e. A -> ( c = (/) \/ c = { (/) } ) )
31	30	orcomd 718	. . . . . . . . . . . . 13 |- ( c e. A -> ( c = { (/) } \/ c = (/) ) )
32	31	3ad2ant3 1004	. . . . . . . . . . . 12 |- ( ( a e. A /\ b e. A /\ c e. A ) -> ( c = { (/) } \/ c = (/) ) )
33	32	adantr 274	. . . . . . . . . . 11 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> ( c = { (/) } \/ c = (/) ) )
34	25, 33	ecased 1327	. . . . . . . . . 10 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> c = { (/) } )
35	7, 21, 34	3eltr3d 2222	. . . . . . . . 9 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> { (/) } e. { (/) } )
36	35	ex 114	. . . . . . . 8 |- ( ( a e. A /\ b e. A /\ c e. A ) -> ( ( a e. b /\ b e. c ) -> { (/) } e. { (/) } ) )
37	6, 36	mtoi 653	. . . . . . 7 |- ( ( a e. A /\ b e. A /\ c e. A ) -> -. ( a e. b /\ b e. c ) )
38	37	adantr 274	. . . . . 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 609	. . . . 5 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> a _E c )
40	4, 39	sylan2b 285	. . . 4 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a _E b /\ b _E c ) ) -> a _E c )
41	40	ex 114	. . 3 |- ( ( a e. A /\ b e. A /\ c e. A ) -> ( ( a _E b /\ b _E c ) -> a _E c ) )
42	41	rgen3 2519	. 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 4256	. 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 926	1 |- _E We A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 697   /\ w3a 962   = wceq 1331   e. wcel 1480   A.wral 2416   {crab 2420   (/)c0 3363   {csn 3527   {cpr 3528   class class class wbr 3929   _E cep 4209   Fr wfr 4250   We wwe 4252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-eprel 4211  df-frfor 4253  df-frind 4254  df-wetr 4256

This theorem is referenced by:  reg3exmid  4494