Theorem reg3exmidlemwe 4536

Description: Lemma for reg3exmid 4537. 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 4531	. 2 |- _E Fr A
2		epel 4251	. . . . . 6 |- ( a _E b <-> a e. b )
3		epel 4251	. . . . . 6 |- ( b _E c <-> b e. c )
4	2, 3	anbi12i 456	. . . . 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 4498	. . . . . . . 8 |- -. { (/) } e. { (/) }
7		simprr 522	. . . . . . . . . 10 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> b e. c )
8		noel 3398	. . . . . . . . . . . . 13 |- -. a e. (/)
9		eleq2 2221	. . . . . . . . . . . . 13 |- ( b = (/) -> ( a e. b <-> a e. (/) ) )
10	8, 9	mtbiri 665	. . . . . . . . . . . 12 |- ( b = (/) -> -. a e. b )
11		simprl 521	. . . . . . . . . . . 12 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> a e. b )
12	10, 11	nsyl3 616	. . . . . . . . . . 11 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> -. b = (/) )
13		elrabi 2865	. . . . . . . . . . . . . . . 16 |- ( b e. { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) } -> b e. { (/) , { (/) } } )
14		reg3exmidlemwe.a	. . . . . . . . . . . . . . . 16 |- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }
15	13, 14	eleq2s 2252	. . . . . . . . . . . . . . 15 |- ( b e. A -> b e. { (/) , { (/) } } )
16		elpri 3583	. . . . . . . . . . . . . . 15 |- ( b e. { (/) , { (/) } } -> ( b = (/) \/ b = { (/) } ) )
17	15, 16	syl 14	. . . . . . . . . . . . . 14 |- ( b e. A -> ( b = (/) \/ b = { (/) } ) )
18	17	orcomd 719	. . . . . . . . . . . . 13 |- ( b e. A -> ( b = { (/) } \/ b = (/) ) )
19	18	3ad2ant2 1004	. . . . . . . . . . . 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 1331	. . . . . . . . . 10 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> b = { (/) } )
22		noel 3398	. . . . . . . . . . . . 13 |- -. b e. (/)
23		eleq2 2221	. . . . . . . . . . . . 13 |- ( c = (/) -> ( b e. c <-> b e. (/) ) )
24	22, 23	mtbiri 665	. . . . . . . . . . . 12 |- ( c = (/) -> -. b e. c )
25	24, 7	nsyl3 616	. . . . . . . . . . 11 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> -. c = (/) )
26		elrabi 2865	. . . . . . . . . . . . . . . 16 |- ( c e. { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) } -> c e. { (/) , { (/) } } )
27	26, 14	eleq2s 2252	. . . . . . . . . . . . . . 15 |- ( c e. A -> c e. { (/) , { (/) } } )
28		vex 2715	. . . . . . . . . . . . . . . 16 |- c e. _V
29	28	elpr 3581	. . . . . . . . . . . . . . 15 |- ( c e. { (/) , { (/) } } <-> ( c = (/) \/ c = { (/) } ) )
30	27, 29	sylib 121	. . . . . . . . . . . . . 14 |- ( c e. A -> ( c = (/) \/ c = { (/) } ) )
31	30	orcomd 719	. . . . . . . . . . . . 13 |- ( c e. A -> ( c = { (/) } \/ c = (/) ) )
32	31	3ad2ant3 1005	. . . . . . . . . . . 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 1331	. . . . . . . . . 10 |- ( ( ( a e. A /\ b e. A /\ c e. A ) /\ ( a e. b /\ b e. c ) ) -> c = { (/) } )
35	7, 21, 34	3eltr3d 2240	. . . . . . . . 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 654	. . . . . . 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 610	. . . . 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 2544	. 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 4293	. 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 927	1 |- _E We A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698   /\ w3a 963   = wceq 1335   e. wcel 2128   A.wral 2435   {crab 2439   (/)c0 3394   {csn 3560   {cpr 3561   class class class wbr 3965   _E cep 4246   Fr wfr 4287   We wwe 4289

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-setind 4494

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-eprel 4248  df-frfor 4290  df-frind 4291  df-wetr 4293

This theorem is referenced by:  reg3exmid  4537