Theorem reg3exmidlemwe 4671

Description: Lemma for reg3exmid 4672. 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 4666	. 2 |- _E Fr A
2		epel 4383	. . . . . 6 |- ( a _E b <-> a e. b )
3		epel 4383	. . . . . 6 |- ( b _E c <-> b e. c )
4	2, 3	anbi12i 460	. . . . 5 |- ( ( a _E b /\ b _E c ) <-> ( a e. b /\ b e. c ) )
5		simpr 110	. . . . . 6 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> ( a e. b /\ b e. c ) )
6		elirr 4633	. . . . . . . 8 |- -. { (/) } e. { (/) }
7		simprr 531	. . . . . . . . . 10 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> b e. c )
8		noel 3495	. . . . . . . . . . . . 13 |- -. a e. (/)
9		eleq2 2293	. . . . . . . . . . . . 13 |- ( b = (/) -> ( a e. b <-> a e. (/) ) )
10	8, 9	mtbiri 679	. . . . . . . . . . . 12 |- ( b = (/) -> -. a e. b )
11		simprl 529	. . . . . . . . . . . 12 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> a e. b )
12	10, 11	nsyl3 629	. . . . . . . . . . 11 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> -. b = (/) )
13		elrabi 2956	. . . . . . . . . . . . . . . 16 |- ( b e. { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) } -> b e. { (/) , { (/) } } )
14		reg3exmidlemwe.a	. . . . . . . . . . . . . . . 16 |- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }
15	13, 14	eleq2s 2324	. . . . . . . . . . . . . . 15 |- ( b e. A -> b e. { (/) , { (/) } } )
16		elpri 3689	. . . . . . . . . . . . . . 15 |- ( b e. { (/) , { (/) } } -> ( b = (/) \/ b = { (/) } ) )
17	15, 16	syl 14	. . . . . . . . . . . . . 14 |- ( b e. A -> ( b = (/) \/ b = { (/) } ) )
18	17	orcomd 734	. . . . . . . . . . . . 13 |- ( b e. A -> ( b = { (/) } \/ b = (/) ) )
19	18	3ad2ant2 1043	. . . . . . . . . . . 12 |- ( ( a e. A /\ b e. A /\ c e. A ) -> ( b = { (/) } \/ b = (/) ) )
20	19	adantr 276	. . . . . . . . . . 11 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> ( b = { (/) } \/ b = (/) ) )
21	12, 20	ecased 1383	. . . . . . . . . 10 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> b = { (/) } )
22		noel 3495	. . . . . . . . . . . . 13 |- -. b e. (/)
23		eleq2 2293	. . . . . . . . . . . . 13 |- ( c = (/) -> ( b e. c <-> b e. (/) ) )
24	22, 23	mtbiri 679	. . . . . . . . . . . 12 |- ( c = (/) -> -. b e. c )
25	24, 7	nsyl3 629	. . . . . . . . . . 11 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> -. c = (/) )
26		elrabi 2956	. . . . . . . . . . . . . . . 16 |- ( c e. { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) } -> c e. { (/) , { (/) } } )
27	26, 14	eleq2s 2324	. . . . . . . . . . . . . . 15 |- ( c e. A -> c e. { (/) , { (/) } } )
28		vex 2802	. . . . . . . . . . . . . . . 16 |- c e. _V
29	28	elpr 3687	. . . . . . . . . . . . . . 15 |- ( c e. { (/) , { (/) } } <-> ( c = (/) \/ c = { (/) } ) )
30	27, 29	sylib 122	. . . . . . . . . . . . . 14 |- ( c e. A -> ( c = (/) \/ c = { (/) } ) )
31	30	orcomd 734	. . . . . . . . . . . . 13 |- ( c e. A -> ( c = { (/) } \/ c = (/) ) )
32	31	3ad2ant3 1044	. . . . . . . . . . . 12 |- ( ( a e. A /\ b e. A /\ c e. A ) -> ( c = { (/) } \/ c = (/) ) )
33	32	adantr 276	. . . . . . . . . . 11 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> ( c = { (/) } \/ c = (/) ) )
34	25, 33	ecased 1383	. . . . . . . . . 10 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> c = { (/) } )
35	7, 21, 34	3eltr3d 2312	. . . . . . . . 9 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> { (/) } e. { (/) } )
36	35	ex 115	. . . . . . . 8 |- ( ( a e. A /\ b e. A /\ c e. A ) -> ( ( a e. b /\ b e. c ) -> { (/) } e. { (/) } ) )
37	6, 36	mtoi 668	. . . . . . 7 |- ( ( a e. A /\ b e. A /\ c e. A ) -> -. ( a e. b /\ b e. c ) )
38	37	adantr 276	. . . . . 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 623	. . . . 5 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> a _E c )
40	4, 39	sylan2b 287	. . . 4 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a _E b /\ b _E c ) ) -> a _E c )
41	40	ex 115	. . 3 |- ( ( a e. A /\ b e. A /\ c e. A ) -> ( ( a _E b /\ b _E c ) -> a _E c ) )
42	41	rgen3 2617	. 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 4425	. 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 948	1 |- _E We A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   {crab 2512   (/)c0 3491   {csn 3666   {cpr 3667   class class class wbr 4083   _E cep 4378   Fr wfr 4419   We wwe 4421

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-eprel 4380  df-frfor 4422  df-frind 4423  df-wetr 4425

This theorem is referenced by:  reg3exmid  4672