Theorem tfinltfin 4502

Description: Ordering rule for the finite T operation. Corollary to theorem X.1.33 of [Rosser] p. 529. (Contributed by SF, 1-Feb-2015.)

Assertion

Ref	Expression
tfinltfin	|- ((M e. Nn /\ N e. Nn ) -> (<<M, N>> e. <fin <-> << Tfin M, Tfin N>> e. <fin ))

Proof of Theorem tfinltfin

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

Step	Hyp	Ref	Expression
1		tfinltfinlem1 4501	. 2 |- ((M e. Nn /\ N e. Nn ) -> (<<M, N>> e. <fin -> << Tfin M, Tfin N>> e. <fin ))
2		tfineq 4489	. . . . . . . . 9 |- (M = (/) -> Tfin M = Tfin (/))
3		tfinnul 4492	. . . . . . . . 9 |- Tfin (/) = (/)
4	2, 3	syl6eq 2401	. . . . . . . 8 |- (M = (/) -> Tfin M = (/))
5		df-ne 2519	. . . . . . . . 9 |- ( Tfin M =/= (/) <-> -. Tfin M = (/))
6	5	con2bii 322	. . . . . . . 8 |- ( Tfin M = (/) <-> -. Tfin M =/= (/))
7	4, 6	sylib 188	. . . . . . 7 |- (M = (/) -> -. Tfin M =/= (/))
8	7	intnanrd 883	. . . . . 6 |- (M = (/) -> -. ( Tfin M =/= (/) /\ E.x e. Nn Tfin N = (( Tfin M +o x) +o 1c)))
9		tfinex 4486	. . . . . . 7 |- Tfin M e. _V
10		tfinex 4486	. . . . . . 7 |- Tfin N e. _V
11		opkltfing 4450	. . . . . . 7 |- (( Tfin M e. _V /\ Tfin N e. _V) -> (<< Tfin M, Tfin N>> e. <fin <-> ( Tfin M =/= (/) /\ E.x e. Nn Tfin N = (( Tfin M +o x) +o 1c))))
12	9, 10, 11	mp2an 653	. . . . . 6 |- (<< Tfin M, Tfin N>> e. <fin <-> ( Tfin M =/= (/) /\ E.x e. Nn Tfin N = (( Tfin M +o x) +o 1c)))
13	8, 12	sylnibr 296	. . . . 5 |- (M = (/) -> -. << Tfin M, Tfin N>> e. <fin )
14	13	pm2.21d 98	. . . 4 |- (M = (/) -> (<< Tfin M, Tfin N>> e. <fin -> <<M, N>> e. <fin ))
15	14	a1d 22	. . 3 |- (M = (/) -> ((M e. Nn /\ N e. Nn ) -> (<< Tfin M, Tfin N>> e. <fin -> <<M, N>> e. <fin )))
16		tfinprop 4490	. . . . . . . . . . . . . 14 |- ((M e. Nn /\ M =/= (/)) -> ( Tfin M e. Nn /\ E.y e. M ~P1 y e. Tfin M))
17	16	simpld 445	. . . . . . . . . . . . 13 |- ((M e. Nn /\ M =/= (/)) -> Tfin M e. Nn )
18		ltfinirr 4458	. . . . . . . . . . . . 13 |- ( Tfin M e. Nn -> -. << Tfin M, Tfin M>> e. <fin )
19	17, 18	syl 15	. . . . . . . . . . . 12 |- ((M e. Nn /\ M =/= (/)) -> -. << Tfin M, Tfin M>> e. <fin )
20	19	3adant2 974	. . . . . . . . . . 11 |- ((M e. Nn /\ N e. Nn /\ M =/= (/)) -> -. << Tfin M, Tfin M>> e. <fin )
21		opkeq2 4061	. . . . . . . . . . . . 13 |- ( Tfin M = Tfin N -> << Tfin M, Tfin M>> = << Tfin M, Tfin N>>)
22	21	eleq1d 2419	. . . . . . . . . . . 12 |- ( Tfin M = Tfin N -> (<< Tfin M, Tfin M>> e. <fin <-> << Tfin M, Tfin N>> e. <fin ))
23	22	notbid 285	. . . . . . . . . . 11 |- ( Tfin M = Tfin N -> (-. << Tfin M, Tfin M>> e. <fin <-> -. << Tfin M, Tfin N>> e. <fin ))
24	20, 23	syl5ibcom 211	. . . . . . . . . 10 |- ((M e. Nn /\ N e. Nn /\ M =/= (/)) -> ( Tfin M = Tfin N -> -. << Tfin M, Tfin N>> e. <fin ))
25	24	con2d 107	. . . . . . . . 9 |- ((M e. Nn /\ N e. Nn /\ M =/= (/)) -> (<< Tfin M, Tfin N>> e. <fin -> -. Tfin M = Tfin N))
26	25	imp 418	. . . . . . . 8 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ << Tfin M, Tfin N>> e. <fin ) -> -. Tfin M = Tfin N)
27		tfineq 4489	. . . . . . . 8 |- (M = N -> Tfin M = Tfin N)
28	26, 27	nsyl 113	. . . . . . 7 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ << Tfin M, Tfin N>> e. <fin ) -> -. M = N)
29		simpl1 958	. . . . . . . . . . . . 13 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ (<< Tfin M, Tfin N>> e. <fin /\ N =/= (/))) -> M e. Nn )
30		simpl3 960	. . . . . . . . . . . . 13 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ (<< Tfin M, Tfin N>> e. <fin /\ N =/= (/))) -> M =/= (/))
31	29, 30, 17	syl2anc 642	. . . . . . . . . . . 12 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ (<< Tfin M, Tfin N>> e. <fin /\ N =/= (/))) -> Tfin M e. Nn )
32		simpl2 959	. . . . . . . . . . . . 13 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ (<< Tfin M, Tfin N>> e. <fin /\ N =/= (/))) -> N e. Nn )
33		simprr 733	. . . . . . . . . . . . 13 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ (<< Tfin M, Tfin N>> e. <fin /\ N =/= (/))) -> N =/= (/))
34		tfinprop 4490	. . . . . . . . . . . . . 14 |- ((N e. Nn /\ N =/= (/)) -> ( Tfin N e. Nn /\ E.y e. N ~P1 y e. Tfin N))
35	34	simpld 445	. . . . . . . . . . . . 13 |- ((N e. Nn /\ N =/= (/)) -> Tfin N e. Nn )
36	32, 33, 35	syl2anc 642	. . . . . . . . . . . 12 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ (<< Tfin M, Tfin N>> e. <fin /\ N =/= (/))) -> Tfin N e. Nn )
37	31, 36	jca 518	. . . . . . . . . . 11 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ (<< Tfin M, Tfin N>> e. <fin /\ N =/= (/))) -> ( Tfin M e. Nn /\ Tfin N e. Nn ))
38		simprl 732	. . . . . . . . . . 11 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ (<< Tfin M, Tfin N>> e. <fin /\ N =/= (/))) -> << Tfin M, Tfin N>> e. <fin )
39		ltfinasym 4461	. . . . . . . . . . 11 |- (( Tfin M e. Nn /\ Tfin N e. Nn ) -> (<< Tfin M, Tfin N>> e. <fin -> -. << Tfin N, Tfin M>> e. <fin ))
40	37, 38, 39	sylc 56	. . . . . . . . . 10 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ (<< Tfin M, Tfin N>> e. <fin /\ N =/= (/))) -> -. << Tfin N, Tfin M>> e. <fin )
41	40	expr 598	. . . . . . . . 9 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ << Tfin M, Tfin N>> e. <fin ) -> (N =/= (/) -> -. << Tfin N, Tfin M>> e. <fin ))
42		imnan 411	. . . . . . . . 9 |- ((N =/= (/) -> -. << Tfin N, Tfin M>> e. <fin ) <-> -. (N =/= (/) /\ << Tfin N, Tfin M>> e. <fin ))
43	41, 42	sylib 188	. . . . . . . 8 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ << Tfin M, Tfin N>> e. <fin ) -> -. (N =/= (/) /\ << Tfin N, Tfin M>> e. <fin ))
44		opkltfing 4450	. . . . . . . . . . . . . 14 |- ((N e. Nn /\ M e. Nn ) -> (<<N, M>> e. <fin <-> (N =/= (/) /\ E.y e. Nn M = ((N +o y) +o 1c))))
45	44	ancoms 439	. . . . . . . . . . . . 13 |- ((M e. Nn /\ N e. Nn ) -> (<<N, M>> e. <fin <-> (N =/= (/) /\ E.y e. Nn M = ((N +o y) +o 1c))))
46	45	3adant3 975	. . . . . . . . . . . 12 |- ((M e. Nn /\ N e. Nn /\ M =/= (/)) -> (<<N, M>> e. <fin <-> (N =/= (/) /\ E.y e. Nn M = ((N +o y) +o 1c))))
47	46	simprbda 606	. . . . . . . . . . 11 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ <<N, M>> e. <fin ) -> N =/= (/))
48	47	adantrl 696	. . . . . . . . . 10 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ (<< Tfin M, Tfin N>> e. <fin /\ <<N, M>> e. <fin )) -> N =/= (/))
49		simpl2 959	. . . . . . . . . . . 12 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ (<< Tfin M, Tfin N>> e. <fin /\ <<N, M>> e. <fin )) -> N e. Nn )
50		simpl1 958	. . . . . . . . . . . 12 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ (<< Tfin M, Tfin N>> e. <fin /\ <<N, M>> e. <fin )) -> M e. Nn )
51	49, 50	jca 518	. . . . . . . . . . 11 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ (<< Tfin M, Tfin N>> e. <fin /\ <<N, M>> e. <fin )) -> (N e. Nn /\ M e. Nn ))
52		simprr 733	. . . . . . . . . . 11 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ (<< Tfin M, Tfin N>> e. <fin /\ <<N, M>> e. <fin )) -> <<N, M>> e. <fin )
53		tfinltfinlem1 4501	. . . . . . . . . . 11 |- ((N e. Nn /\ M e. Nn ) -> (<<N, M>> e. <fin -> << Tfin N, Tfin M>> e. <fin ))
54	51, 52, 53	sylc 56	. . . . . . . . . 10 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ (<< Tfin M, Tfin N>> e. <fin /\ <<N, M>> e. <fin )) -> << Tfin N, Tfin M>> e. <fin )
55	48, 54	jca 518	. . . . . . . . 9 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ (<< Tfin M, Tfin N>> e. <fin /\ <<N, M>> e. <fin )) -> (N =/= (/) /\ << Tfin N, Tfin M>> e. <fin ))
56	55	expr 598	. . . . . . . 8 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ << Tfin M, Tfin N>> e. <fin ) -> (<<N, M>> e. <fin -> (N =/= (/) /\ << Tfin N, Tfin M>> e. <fin )))
57	43, 56	mtod 168	. . . . . . 7 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ << Tfin M, Tfin N>> e. <fin ) -> -. <<N, M>> e. <fin )
58		ltfintri 4467	. . . . . . . 8 |- ((M e. Nn /\ N e. Nn /\ M =/= (/)) -> (<<M, N>> e. <fin \/ M = N \/ <<N, M>> e. <fin ))
59	58	adantr 451	. . . . . . 7 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ << Tfin M, Tfin N>> e. <fin ) -> (<<M, N>> e. <fin \/ M = N \/ <<N, M>> e. <fin ))
60	28, 57, 59	ecase23d 1285	. . . . . 6 |- (((M e. Nn /\ N e. Nn /\ M =/= (/)) /\ << Tfin M, Tfin N>> e. <fin ) -> <<M, N>> e. <fin )
61	60	ex 423	. . . . 5 |- ((M e. Nn /\ N e. Nn /\ M =/= (/)) -> (<< Tfin M, Tfin N>> e. <fin -> <<M, N>> e. <fin ))
62	61	3expa 1151	. . . 4 |- (((M e. Nn /\ N e. Nn ) /\ M =/= (/)) -> (<< Tfin M, Tfin N>> e. <fin -> <<M, N>> e. <fin ))
63	62	expcom 424	. . 3 |- (M =/= (/) -> ((M e. Nn /\ N e. Nn ) -> (<< Tfin M, Tfin N>> e. <fin -> <<M, N>> e. <fin )))
64	15, 63	pm2.61ine 2593	. 2 |- ((M e. Nn /\ N e. Nn ) -> (<< Tfin M, Tfin N>> e. <fin -> <<M, N>> e. <fin ))
65	1, 64	impbid 183	1 |- ((M e. Nn /\ N e. Nn ) -> (<<M, N>> e. <fin <-> << Tfin M, Tfin N>> e. <fin ))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358   \/ w3o 933   /\ w3a 934   = wceq 1642   e. wcel 1710   =/= wne 2517  E.wrex 2616  _Vcvv 2860  (/)c0 3551  <<copk 4058  1_cc1c 4135  ~P₁ cpw1 4136   Nn cnnc 4374   +o cplc 4376   <_fin cltfin 4434   T_fin ctfin 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-lefin 4441  df-ltfin 4442  df-tfin 4444

This theorem is referenced by:  tfinlefin  4503  sfinltfin  4536