Theorem noinfep 8557

Description: Using the Axiom of Regularity in the form zfregfr 8509, show that there are no infinite descending e.-chains. Proposition 7.34 of [TakeutiZaring] p. 44. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 22-Mar-2013.)

Assertion

Ref	Expression
noinfep	|- E. x e. om ( F ` suc x ) e/ ( F ` x )

Distinct variable group:   x, F

Proof of Theorem noinfep

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

Step	Hyp	Ref	Expression
1		omex 8540	. . . . 5 |- om e. _V
2	1	mptex 6486	. . . 4 |- ( w e. om |-> ( F ` w ) ) e. _V
3	2	rnex 7100	. . 3 |- ran ( w e. om |-> ( F ` w ) ) e. _V
4		zfregfr 8509	. . 3 |- _E Fr ran ( w e. om |-> ( F ` w ) )
5		ssid 3624	. . 3 |- ran ( w e. om |-> ( F ` w ) ) C_ ran ( w e. om |-> ( F ` w ) )
6		dmmptg 5632	. . . . . 6 |- ( A. w e. om ( F ` w ) e. _V -> dom ( w e. om |-> ( F ` w ) ) = om )
7		fvexd 6203	. . . . . 6 |- ( w e. om -> ( F ` w ) e. _V )
8	6, 7	mprg 2926	. . . . 5 |- dom ( w e. om |-> ( F ` w ) ) = om
9		peano1 7085	. . . . . 6 |- (/) e. om
10	9	ne0ii 3923	. . . . 5 |- om =/= (/)
11	8, 10	eqnetri 2864	. . . 4 |- dom ( w e. om |-> ( F ` w ) ) =/= (/)
12		dm0rn0 5342	. . . . 5 |- ( dom ( w e. om |-> ( F ` w ) ) = (/) <-> ran ( w e. om |-> ( F ` w ) ) = (/) )
13	12	necon3bii 2846	. . . 4 |- ( dom ( w e. om |-> ( F ` w ) ) =/= (/) <-> ran ( w e. om |-> ( F ` w ) ) =/= (/) )
14	11, 13	mpbi 220	. . 3 |- ran ( w e. om |-> ( F ` w ) ) =/= (/)
15		fri 5076	. . 3 |- ( ( ( ran ( w e. om |-> ( F ` w ) ) e. _V /\ _E Fr ran ( w e. om |-> ( F ` w ) ) ) /\ ( ran ( w e. om |-> ( F ` w ) ) C_ ran ( w e. om |-> ( F ` w ) ) /\ ran ( w e. om |-> ( F ` w ) ) =/= (/) ) ) -> E. y e. ran ( w e. om |-> ( F ` w ) ) A. z e. ran ( w e. om |-> ( F ` w ) ) -. z _E y )
16	3, 4, 5, 14, 15	mp4an 709	. 2 |- E. y e. ran ( w e. om |-> ( F ` w ) ) A. z e. ran ( w e. om |-> ( F ` w ) ) -. z _E y
17		fvex 6201	. . . . . . 7 |- ( F ` w ) e. _V
18		eqid 2622	. . . . . . 7 |- ( w e. om |-> ( F ` w ) ) = ( w e. om |-> ( F ` w ) )
19	17, 18	fnmpti 6022	. . . . . 6 |- ( w e. om |-> ( F ` w ) ) Fn om
20		fvelrnb 6243	. . . . . 6 |- ( ( w e. om |-> ( F ` w ) ) Fn om -> ( y e. ran ( w e. om |-> ( F ` w ) ) <-> E. x e. om ( ( w e. om |-> ( F ` w ) ) ` x ) = y ) )
21	19, 20	ax-mp 5	. . . . 5 |- ( y e. ran ( w e. om |-> ( F ` w ) ) <-> E. x e. om ( ( w e. om |-> ( F ` w ) ) ` x ) = y )
22		peano2 7086	. . . . . . . . . 10 |- ( x e. om -> suc x e. om )
23		fveq2 6191	. . . . . . . . . . 11 |- ( w = suc x -> ( F ` w ) = ( F ` suc x ) )
24		fvex 6201	. . . . . . . . . . 11 |- ( F ` suc x ) e. _V
25	23, 18, 24	fvmpt 6282	. . . . . . . . . 10 |- ( suc x e. om -> ( ( w e. om |-> ( F ` w ) ) ` suc x ) = ( F ` suc x ) )
26	22, 25	syl 17	. . . . . . . . 9 |- ( x e. om -> ( ( w e. om |-> ( F ` w ) ) ` suc x ) = ( F ` suc x ) )
27		fnfvelrn 6356	. . . . . . . . . 10 |- ( ( ( w e. om |-> ( F ` w ) ) Fn om /\ suc x e. om ) -> ( ( w e. om |-> ( F ` w ) ) ` suc x ) e. ran ( w e. om |-> ( F ` w ) ) )
28	19, 22, 27	sylancr 695	. . . . . . . . 9 |- ( x e. om -> ( ( w e. om |-> ( F ` w ) ) ` suc x ) e. ran ( w e. om |-> ( F ` w ) ) )
29	26, 28	eqeltrrd 2702	. . . . . . . 8 |- ( x e. om -> ( F ` suc x ) e. ran ( w e. om |-> ( F ` w ) ) )
30		epel 5032	. . . . . . . . . . . 12 |- ( z _E y <-> z e. y )
31		eleq1 2689	. . . . . . . . . . . 12 |- ( z = ( F ` suc x ) -> ( z e. y <-> ( F ` suc x ) e. y ) )
32	30, 31	syl5bb 272	. . . . . . . . . . 11 |- ( z = ( F ` suc x ) -> ( z _E y <-> ( F ` suc x ) e. y ) )
33	32	notbid 308	. . . . . . . . . 10 |- ( z = ( F ` suc x ) -> ( -. z _E y <-> -. ( F ` suc x ) e. y ) )
34		df-nel 2898	. . . . . . . . . 10 |- ( ( F ` suc x ) e/ y <-> -. ( F ` suc x ) e. y )
35	33, 34	syl6bbr 278	. . . . . . . . 9 |- ( z = ( F ` suc x ) -> ( -. z _E y <-> ( F ` suc x ) e/ y ) )
36	35	rspccv 3306	. . . . . . . 8 |- ( A. z e. ran ( w e. om |-> ( F ` w ) ) -. z _E y -> ( ( F ` suc x ) e. ran ( w e. om |-> ( F ` w ) ) -> ( F ` suc x ) e/ y ) )
37	29, 36	syl5com 31	. . . . . . 7 |- ( x e. om -> ( A. z e. ran ( w e. om |-> ( F ` w ) ) -. z _E y -> ( F ` suc x ) e/ y ) )
38		fveq2 6191	. . . . . . . . . . 11 |- ( w = x -> ( F ` w ) = ( F ` x ) )
39		fvex 6201	. . . . . . . . . . 11 |- ( F ` x ) e. _V
40	38, 18, 39	fvmpt 6282	. . . . . . . . . 10 |- ( x e. om -> ( ( w e. om |-> ( F ` w ) ) ` x ) = ( F ` x ) )
41		eqeq1 2626	. . . . . . . . . 10 |- ( ( ( w e. om |-> ( F ` w ) ) ` x ) = y -> ( ( ( w e. om |-> ( F ` w ) ) ` x ) = ( F ` x ) <-> y = ( F ` x ) ) )
42	40, 41	syl5ibcom 235	. . . . . . . . 9 |- ( x e. om -> ( ( ( w e. om |-> ( F ` w ) ) ` x ) = y -> y = ( F ` x ) ) )
43		neleq2 2903	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( ( F ` suc x ) e/ y <-> ( F ` suc x ) e/ ( F ` x ) ) )
44	43	biimpd 219	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( ( F ` suc x ) e/ y -> ( F ` suc x ) e/ ( F ` x ) ) )
45	42, 44	syl6 35	. . . . . . . 8 |- ( x e. om -> ( ( ( w e. om |-> ( F ` w ) ) ` x ) = y -> ( ( F ` suc x ) e/ y -> ( F ` suc x ) e/ ( F ` x ) ) ) )
46	45	com23 86	. . . . . . 7 |- ( x e. om -> ( ( F ` suc x ) e/ y -> ( ( ( w e. om |-> ( F ` w ) ) ` x ) = y -> ( F ` suc x ) e/ ( F ` x ) ) ) )
47	37, 46	syldc 48	. . . . . 6 |- ( A. z e. ran ( w e. om |-> ( F ` w ) ) -. z _E y -> ( x e. om -> ( ( ( w e. om |-> ( F ` w ) ) ` x ) = y -> ( F ` suc x ) e/ ( F ` x ) ) ) )
48	47	reximdvai 3015	. . . . 5 |- ( A. z e. ran ( w e. om |-> ( F ` w ) ) -. z _E y -> ( E. x e. om ( ( w e. om |-> ( F ` w ) ) ` x ) = y -> E. x e. om ( F ` suc x ) e/ ( F ` x ) ) )
49	21, 48	syl5bi 232	. . . 4 |- ( A. z e. ran ( w e. om |-> ( F ` w ) ) -. z _E y -> ( y e. ran ( w e. om |-> ( F ` w ) ) -> E. x e. om ( F ` suc x ) e/ ( F ` x ) ) )
50	49	com12 32	. . 3 |- ( y e. ran ( w e. om |-> ( F ` w ) ) -> ( A. z e. ran ( w e. om |-> ( F ` w ) ) -. z _E y -> E. x e. om ( F ` suc x ) e/ ( F ` x ) ) )
51	50	rexlimiv 3027	. 2 |- ( E. y e. ran ( w e. om |-> ( F ` w ) ) A. z e. ran ( w e. om |-> ( F ` w ) ) -. z _E y -> E. x e. om ( F ` suc x ) e/ ( F ` x ) )
52	16, 51	ax-mp 5	1 |- E. x e. om ( F ` suc x ) e/ ( F ` x )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729   _E cep 5028   Fr wfr 5070   dom cdm 5114   ran crn 5115   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065

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-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  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-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-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-om 7066

This theorem is referenced by: (None)