Theorem wofib 8450

Description: The only sets which are well-ordered forwards and backwards are finite sets. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 23-May-2015.)

Hypothesis

Ref	Expression
wofib.1	|- A e. _V

Assertion

Ref	Expression
wofib	|- ( ( R Or A /\ A e. Fin ) <-> ( R We A /\ `' R We A ) )

Proof of Theorem wofib

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wofi 8209	. . 3 |- ( ( R Or A /\ A e. Fin ) -> R We A )
2		cnvso 5674	. . . 4 |- ( R Or A <-> `' R Or A )
3		wofi 8209	. . . 4 |- ( ( `' R Or A /\ A e. Fin ) -> `' R We A )
4	2, 3	sylanb 489	. . 3 |- ( ( R Or A /\ A e. Fin ) -> `' R We A )
5	1, 4	jca 554	. 2 |- ( ( R Or A /\ A e. Fin ) -> ( R We A /\ `' R We A ) )
6		weso 5105	. . . 4 |- ( R We A -> R Or A )
7	6	adantr 481	. . 3 |- ( ( R We A /\ `' R We A ) -> R Or A )
8		peano2 7086	. . . . . . . . 9 |- ( y e. om -> suc y e. om )
9		sucidg 5803	. . . . . . . . 9 |- ( y e. om -> y e. suc y )
10		vex 3203	. . . . . . . . . . . . 13 |- z e. _V
11		vex 3203	. . . . . . . . . . . . 13 |- y e. _V
12	10, 11	brcnv 5305	. . . . . . . . . . . 12 |- ( z `' _E y <-> y _E z )
13		epel 5032	. . . . . . . . . . . 12 |- ( y _E z <-> y e. z )
14	12, 13	bitri 264	. . . . . . . . . . 11 |- ( z `' _E y <-> y e. z )
15		eleq2 2690	. . . . . . . . . . 11 |- ( z = suc y -> ( y e. z <-> y e. suc y ) )
16	14, 15	syl5bb 272	. . . . . . . . . 10 |- ( z = suc y -> ( z `' _E y <-> y e. suc y ) )
17	16	rspcev 3309	. . . . . . . . 9 |- ( ( suc y e. om /\ y e. suc y ) -> E. z e. om z `' _E y )
18	8, 9, 17	syl2anc 693	. . . . . . . 8 |- ( y e. om -> E. z e. om z `' _E y )
19		dfrex2 2996	. . . . . . . 8 |- ( E. z e. om z `' _E y <-> -. A. z e. om -. z `' _E y )
20	18, 19	sylib 208	. . . . . . 7 |- ( y e. om -> -. A. z e. om -. z `' _E y )
21	20	nrex 3000	. . . . . 6 |- -. E. y e. om A. z e. om -. z `' _E y
22		ordom 7074	. . . . . . . 8 |- Ord om
23		eqid 2622	. . . . . . . . 9 |- OrdIso ( R , A ) = OrdIso ( R , A )
24	23	oicl 8434	. . . . . . . 8 |- Ord dom OrdIso ( R , A )
25		ordtri1 5756	. . . . . . . 8 |- ( ( Ord om /\ Ord dom OrdIso ( R , A ) ) -> ( om C_ dom OrdIso ( R , A ) <-> -. dom OrdIso ( R , A ) e. om ) )
26	22, 24, 25	mp2an 708	. . . . . . 7 |- ( om C_ dom OrdIso ( R , A ) <-> -. dom OrdIso ( R , A ) e. om )
27		wofib.1	. . . . . . . . . . 11 |- A e. _V
28	23	oion 8441	. . . . . . . . . . 11 |- ( A e. _V -> dom OrdIso ( R , A ) e. On )
29	27, 28	mp1i 13	. . . . . . . . . 10 |- ( ( ( R We A /\ `' R We A ) /\ om C_ dom OrdIso ( R , A ) ) -> dom OrdIso ( R , A ) e. On )
30		simpr 477	. . . . . . . . . 10 |- ( ( ( R We A /\ `' R We A ) /\ om C_ dom OrdIso ( R , A ) ) -> om C_ dom OrdIso ( R , A ) )
31	29, 30	ssexd 4805	. . . . . . . . 9 |- ( ( ( R We A /\ `' R We A ) /\ om C_ dom OrdIso ( R , A ) ) -> om e. _V )
32	23	oiiso 8442	. . . . . . . . . . . . 13 |- ( ( A e. _V /\ R We A ) -> OrdIso ( R , A ) Isom _E , R ( dom OrdIso ( R , A ) , A ) )
33	27, 32	mpan 706	. . . . . . . . . . . 12 |- ( R We A -> OrdIso ( R , A ) Isom _E , R ( dom OrdIso ( R , A ) , A ) )
34		isocnv2 6581	. . . . . . . . . . . 12 |- (OrdIso ( R , A ) Isom _E , R ( dom OrdIso ( R , A ) , A ) <-> OrdIso ( R , A ) Isom `' _E , `' R ( dom OrdIso ( R , A ) , A ) )
35	33, 34	sylib 208	. . . . . . . . . . 11 |- ( R We A -> OrdIso ( R , A ) Isom `' _E , `' R ( dom OrdIso ( R , A ) , A ) )
36		wefr 5104	. . . . . . . . . . 11 |- ( `' R We A -> `' R Fr A )
37		isofr 6592	. . . . . . . . . . . 12 |- (OrdIso ( R , A ) Isom `' _E , `' R ( dom OrdIso ( R , A ) , A ) -> ( `' _E Fr dom OrdIso ( R , A ) <-> `' R Fr A ) )
38	37	biimpar 502	. . . . . . . . . . 11 |- ( (OrdIso ( R , A ) Isom `' _E , `' R ( dom OrdIso ( R , A ) , A ) /\ `' R Fr A ) -> `' _E Fr dom OrdIso ( R , A ) )
39	35, 36, 38	syl2an 494	. . . . . . . . . 10 |- ( ( R We A /\ `' R We A ) -> `' _E Fr dom OrdIso ( R , A ) )
40	39	adantr 481	. . . . . . . . 9 |- ( ( ( R We A /\ `' R We A ) /\ om C_ dom OrdIso ( R , A ) ) -> `' _E Fr dom OrdIso ( R , A ) )
41		1onn 7719	. . . . . . . . . 10 |- 1o e. om
42		ne0i 3921	. . . . . . . . . 10 |- ( 1o e. om -> om =/= (/) )
43	41, 42	mp1i 13	. . . . . . . . 9 |- ( ( ( R We A /\ `' R We A ) /\ om C_ dom OrdIso ( R , A ) ) -> om =/= (/) )
44		fri 5076	. . . . . . . . 9 |- ( ( ( om e. _V /\ `' _E Fr dom OrdIso ( R , A ) ) /\ ( om C_ dom OrdIso ( R , A ) /\ om =/= (/) ) ) -> E. y e. om A. z e. om -. z `' _E y )
45	31, 40, 30, 43, 44	syl22anc 1327	. . . . . . . 8 |- ( ( ( R We A /\ `' R We A ) /\ om C_ dom OrdIso ( R , A ) ) -> E. y e. om A. z e. om -. z `' _E y )
46	45	ex 450	. . . . . . 7 |- ( ( R We A /\ `' R We A ) -> ( om C_ dom OrdIso ( R , A ) -> E. y e. om A. z e. om -. z `' _E y ) )
47	26, 46	syl5bir 233	. . . . . 6 |- ( ( R We A /\ `' R We A ) -> ( -. dom OrdIso ( R , A ) e. om -> E. y e. om A. z e. om -. z `' _E y ) )
48	21, 47	mt3i 141	. . . . 5 |- ( ( R We A /\ `' R We A ) -> dom OrdIso ( R , A ) e. om )
49		ssid 3624	. . . . 5 |- dom OrdIso ( R , A ) C_ dom OrdIso ( R , A )
50		ssnnfi 8179	. . . . 5 |- ( ( dom OrdIso ( R , A ) e. om /\ dom OrdIso ( R , A ) C_ dom OrdIso ( R , A ) ) -> dom OrdIso ( R , A ) e. Fin )
51	48, 49, 50	sylancl 694	. . . 4 |- ( ( R We A /\ `' R We A ) -> dom OrdIso ( R , A ) e. Fin )
52		simpl 473	. . . . . 6 |- ( ( R We A /\ `' R We A ) -> R We A )
53	23	oien 8443	. . . . . 6 |- ( ( A e. _V /\ R We A ) -> dom OrdIso ( R , A ) ~~ A )
54	27, 52, 53	sylancr 695	. . . . 5 |- ( ( R We A /\ `' R We A ) -> dom OrdIso ( R , A ) ~~ A )
55		enfi 8176	. . . . 5 |- ( dom OrdIso ( R , A ) ~~ A -> ( dom OrdIso ( R , A ) e. Fin <-> A e. Fin ) )
56	54, 55	syl 17	. . . 4 |- ( ( R We A /\ `' R We A ) -> ( dom OrdIso ( R , A ) e. Fin <-> A e. Fin ) )
57	51, 56	mpbid 222	. . 3 |- ( ( R We A /\ `' R We A ) -> A e. Fin )
58	7, 57	jca 554	. 2 |- ( ( R We A /\ `' R We A ) -> ( R Or A /\ A e. Fin ) )
59	5, 58	impbii 199	1 |- ( ( R Or A /\ A e. Fin ) <-> ( R We A /\ `' R We A ) )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653   _E cep 5028   Or wor 5034   Fr wfr 5070   We wwe 5072   `'ccnv 5113   dom cdm 5114   Ord word 5722   Oncon0 5723   suc csuc 5725   Isom wiso 5889   omcom 7065   1oc1o 7553   ~~ cen 7952   Fincfn 7955  OrdIsocoi 8414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-wrecs 7407  df-recs 7468  df-1o 7560  df-er 7742  df-en 7956  df-fin 7959  df-oi 8415

This theorem is referenced by: (None)