Theorem tfin11 4494

Description: The finite T operator is one-to-one over the naturals. Theorem X.1.30 of [Rosser] p. 528. (Contributed by SF, 24-Jan-2015.)

Assertion

Ref	Expression
tfin11	|- ((M e. Nn /\ N e. Nn /\ Tfin M = Tfin N) -> M = N)

Proof of Theorem tfin11

Dummy variables a b p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfinnnul 4491	. . . . . . . 8 |- ((M e. Nn /\ M =/= (/)) -> Tfin M =/= (/))
2	1	ex 423	. . . . . . 7 |- (M e. Nn -> (M =/= (/) -> Tfin M =/= (/)))
3	2	necon4d 2580	. . . . . 6 |- (M e. Nn -> ( Tfin M = (/) -> M = (/)))
4	3	3ad2ant1 976	. . . . 5 |- ((M e. Nn /\ N e. Nn /\ Tfin M = Tfin N) -> ( Tfin M = (/) -> M = (/)))
5	4	impcom 419	. . . 4 |- (( Tfin M = (/) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) -> M = (/))
6		eqeq1 2359	. . . . . . . 8 |- ( Tfin M = Tfin N -> ( Tfin M = (/) <-> Tfin N = (/)))
7	6	adantl 452	. . . . . . 7 |- ((N e. Nn /\ Tfin M = Tfin N) -> ( Tfin M = (/) <-> Tfin N = (/)))
8		tfinnnul 4491	. . . . . . . . . 10 |- ((N e. Nn /\ N =/= (/)) -> Tfin N =/= (/))
9	8	ex 423	. . . . . . . . 9 |- (N e. Nn -> (N =/= (/) -> Tfin N =/= (/)))
10	9	necon4d 2580	. . . . . . . 8 |- (N e. Nn -> ( Tfin N = (/) -> N = (/)))
11	10	adantr 451	. . . . . . 7 |- ((N e. Nn /\ Tfin M = Tfin N) -> ( Tfin N = (/) -> N = (/)))
12	7, 11	sylbid 206	. . . . . 6 |- ((N e. Nn /\ Tfin M = Tfin N) -> ( Tfin M = (/) -> N = (/)))
13	12	3adant1 973	. . . . 5 |- ((M e. Nn /\ N e. Nn /\ Tfin M = Tfin N) -> ( Tfin M = (/) -> N = (/)))
14	13	impcom 419	. . . 4 |- (( Tfin M = (/) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) -> N = (/))
15	5, 14	eqtr4d 2388	. . 3 |- (( Tfin M = (/) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) -> M = N)
16	15	ex 423	. 2 |- ( Tfin M = (/) -> ((M e. Nn /\ N e. Nn /\ Tfin M = Tfin N) -> M = N))
17		neeq1 2525	. . . . . . . 8 |- ( Tfin M = Tfin N -> ( Tfin M =/= (/) <-> Tfin N =/= (/)))
18	17	biimpd 198	. . . . . . 7 |- ( Tfin M = Tfin N -> ( Tfin M =/= (/) -> Tfin N =/= (/)))
19	18	ancld 536	. . . . . 6 |- ( Tfin M = Tfin N -> ( Tfin M =/= (/) -> ( Tfin M =/= (/) /\ Tfin N =/= (/))))
20		tfineq 4489	. . . . . . . . 9 |- (M = (/) -> Tfin M = Tfin (/))
21		tfinnul 4492	. . . . . . . . 9 |- Tfin (/) = (/)
22	20, 21	syl6eq 2401	. . . . . . . 8 |- (M = (/) -> Tfin M = (/))
23	22	necon3i 2556	. . . . . . 7 |- ( Tfin M =/= (/) -> M =/= (/))
24		tfineq 4489	. . . . . . . . 9 |- (N = (/) -> Tfin N = Tfin (/))
25	24, 21	syl6eq 2401	. . . . . . . 8 |- (N = (/) -> Tfin N = (/))
26	25	necon3i 2556	. . . . . . 7 |- ( Tfin N =/= (/) -> N =/= (/))
27	23, 26	anim12i 549	. . . . . 6 |- (( Tfin M =/= (/) /\ Tfin N =/= (/)) -> (M =/= (/) /\ N =/= (/)))
28	19, 27	syl6 29	. . . . 5 |- ( Tfin M = Tfin N -> ( Tfin M =/= (/) -> (M =/= (/) /\ N =/= (/))))
29	28	3ad2ant3 978	. . . 4 |- ((M e. Nn /\ N e. Nn /\ Tfin M = Tfin N) -> ( Tfin M =/= (/) -> (M =/= (/) /\ N =/= (/))))
30		tfinprop 4490	. . . . . . 7 |- ((M e. Nn /\ M =/= (/)) -> ( Tfin M e. Nn /\ E.a e. M ~P1 a e. Tfin M))
31	30	ex 423	. . . . . 6 |- (M e. Nn -> (M =/= (/) -> ( Tfin M e. Nn /\ E.a e. M ~P1 a e. Tfin M)))
32	31	3ad2ant1 976	. . . . 5 |- ((M e. Nn /\ N e. Nn /\ Tfin M = Tfin N) -> (M =/= (/) -> ( Tfin M e. Nn /\ E.a e. M ~P1 a e. Tfin M)))
33		tfinprop 4490	. . . . . . 7 |- ((N e. Nn /\ N =/= (/)) -> ( Tfin N e. Nn /\ E.b e. N ~P1 b e. Tfin N))
34	33	ex 423	. . . . . 6 |- (N e. Nn -> (N =/= (/) -> ( Tfin N e. Nn /\ E.b e. N ~P1 b e. Tfin N)))
35	34	3ad2ant2 977	. . . . 5 |- ((M e. Nn /\ N e. Nn /\ Tfin M = Tfin N) -> (N =/= (/) -> ( Tfin N e. Nn /\ E.b e. N ~P1 b e. Tfin N)))
36		reeanv 2779	. . . . . . . 8 |- (E.a e. M E.b e. N (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) <-> (E.a e. M ~P1 a e. Tfin M /\ E.b e. N ~P1 b e. Tfin N))
37		simp31 991	. . . . . . . . . . . . 13 |- (((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) -> M e. Nn )
38		tfincl 4493	. . . . . . . . . . . . 13 |- (M e. Nn -> Tfin M e. Nn )
39	37, 38	syl 15	. . . . . . . . . . . 12 |- (((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) -> Tfin M e. Nn )
40		simp2l 981	. . . . . . . . . . . 12 |- (((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) -> ~P1 a e. Tfin M)
41		simp2r 982	. . . . . . . . . . . . 13 |- (((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) -> ~P1 b e. Tfin N)
42		simp33 993	. . . . . . . . . . . . 13 |- (((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) -> Tfin M = Tfin N)
43	41, 42	eleqtrrd 2430	. . . . . . . . . . . 12 |- (((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) -> ~P1 b e. Tfin M)
44		ncfinlower 4484	. . . . . . . . . . . 12 |- (( Tfin M e. Nn /\ ~P1 a e. Tfin M /\ ~P1 b e. Tfin M) -> E.p e. Nn (a e. p /\ b e. p))
45	39, 40, 43, 44	syl3anc 1182	. . . . . . . . . . 11 |- (((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) -> E.p e. Nn (a e. p /\ b e. p))
46		simpl31 1036	. . . . . . . . . . . . . . 15 |- ((((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) /\ (p e. Nn /\ (a e. p /\ b e. p))) -> M e. Nn )
47		simprl 732	. . . . . . . . . . . . . . 15 |- ((((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) /\ (p e. Nn /\ (a e. p /\ b e. p))) -> p e. Nn )
48		simpl1l 1006	. . . . . . . . . . . . . . 15 |- ((((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) /\ (p e. Nn /\ (a e. p /\ b e. p))) -> a e. M)
49		simprrl 740	. . . . . . . . . . . . . . 15 |- ((((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) /\ (p e. Nn /\ (a e. p /\ b e. p))) -> a e. p)
50		nnceleq 4431	. . . . . . . . . . . . . . 15 |- (((M e. Nn /\ p e. Nn ) /\ (a e. M /\ a e. p)) -> M = p)
51	46, 47, 48, 49, 50	syl22anc 1183	. . . . . . . . . . . . . 14 |- ((((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) /\ (p e. Nn /\ (a e. p /\ b e. p))) -> M = p)
52		simpl32 1037	. . . . . . . . . . . . . . 15 |- ((((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) /\ (p e. Nn /\ (a e. p /\ b e. p))) -> N e. Nn )
53		simpl1r 1007	. . . . . . . . . . . . . . 15 |- ((((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) /\ (p e. Nn /\ (a e. p /\ b e. p))) -> b e. N)
54		simprrr 741	. . . . . . . . . . . . . . 15 |- ((((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) /\ (p e. Nn /\ (a e. p /\ b e. p))) -> b e. p)
55		nnceleq 4431	. . . . . . . . . . . . . . 15 |- (((N e. Nn /\ p e. Nn ) /\ (b e. N /\ b e. p)) -> N = p)
56	52, 47, 53, 54, 55	syl22anc 1183	. . . . . . . . . . . . . 14 |- ((((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) /\ (p e. Nn /\ (a e. p /\ b e. p))) -> N = p)
57	51, 56	eqtr4d 2388	. . . . . . . . . . . . 13 |- ((((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) /\ (p e. Nn /\ (a e. p /\ b e. p))) -> M = N)
58	57	expr 598	. . . . . . . . . . . 12 |- ((((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) /\ p e. Nn ) -> ((a e. p /\ b e. p) -> M = N))
59	58	rexlimdva 2739	. . . . . . . . . . 11 |- (((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) -> (E.p e. Nn (a e. p /\ b e. p) -> M = N))
60	45, 59	mpd 14	. . . . . . . . . 10 |- (((a e. M /\ b e. N) /\ (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) /\ (M e. Nn /\ N e. Nn /\ Tfin M = Tfin N)) -> M = N)
61	60	3exp 1150	. . . . . . . . 9 |- ((a e. M /\ b e. N) -> ((~P1 a e. Tfin M /\ ~P1 b e. Tfin N) -> ((M e. Nn /\ N e. Nn /\ Tfin M = Tfin N) -> M = N)))
62	61	rexlimivv 2744	. . . . . . . 8 |- (E.a e. M E.b e. N (~P1 a e. Tfin M /\ ~P1 b e. Tfin N) -> ((M e. Nn /\ N e. Nn /\ Tfin M = Tfin N) -> M = N))
63	36, 62	sylbir 204	. . . . . . 7 |- ((E.a e. M ~P1 a e. Tfin M /\ E.b e. N ~P1 b e. Tfin N) -> ((M e. Nn /\ N e. Nn /\ Tfin M = Tfin N) -> M = N))
64	63	ad2ant2l 726	. . . . . 6 |- ((( Tfin M e. Nn /\ E.a e. M ~P1 a e. Tfin M) /\ ( Tfin N e. Nn /\ E.b e. N ~P1 b e. Tfin N)) -> ((M e. Nn /\ N e. Nn /\ Tfin M = Tfin N) -> M = N))
65	64	com12 27	. . . . 5 |- ((M e. Nn /\ N e. Nn /\ Tfin M = Tfin N) -> ((( Tfin M e. Nn /\ E.a e. M ~P1 a e. Tfin M) /\ ( Tfin N e. Nn /\ E.b e. N ~P1 b e. Tfin N)) -> M = N))
66	32, 35, 65	syl2and 469	. . . 4 |- ((M e. Nn /\ N e. Nn /\ Tfin M = Tfin N) -> ((M =/= (/) /\ N =/= (/)) -> M = N))
67	29, 66	syld 40	. . 3 |- ((M e. Nn /\ N e. Nn /\ Tfin M = Tfin N) -> ( Tfin M =/= (/) -> M = N))
68	67	com12 27	. 2 |- ( Tfin M =/= (/) -> ((M e. Nn /\ N e. Nn /\ Tfin M = Tfin N) -> M = N))
69	16, 68	pm2.61ine 2593	1 |- ((M e. Nn /\ N e. Nn /\ Tfin M = Tfin N) -> M = N)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710   =/= wne 2517  E.wrex 2616  (/)c0 3551  ~P₁ cpw1 4136   Nn cnnc 4374   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-13 1712  ax-14 1714  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-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-tfin 4444

This theorem is referenced by:  tfinnn  4535  vfinspsslem1  4551