Theorem inf3lem1 8525

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8532 for detailed description. (Contributed by NM, 28-Oct-1996.)

Hypotheses

Ref	Expression
inf3lem.1	|- G = ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } )
inf3lem.2	|- F = ( rec ( G , (/) ) |` om )
inf3lem.3	|- A e. _V
inf3lem.4	|- B e. _V

Assertion

Ref	Expression
inf3lem1	|- ( A e. om -> ( F ` A ) C_ ( F ` suc A ) )

Distinct variable group:   x, y, w

Allowed substitution hints:   A( x, y, w)   B( x, y, w)   F( x, y, w)   G( x, y, w)

Proof of Theorem inf3lem1

Dummy variables v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . 3 |- ( v = (/) -> ( F ` v ) = ( F ` (/) ) )
2		suceq 5790	. . . 4 |- ( v = (/) -> suc v = suc (/) )
3	2	fveq2d 6195	. . 3 |- ( v = (/) -> ( F ` suc v ) = ( F ` suc (/) ) )
4	1, 3	sseq12d 3634	. 2 |- ( v = (/) -> ( ( F ` v ) C_ ( F ` suc v ) <-> ( F ` (/) ) C_ ( F ` suc (/) ) ) )
5		fveq2 6191	. . 3 |- ( v = u -> ( F ` v ) = ( F ` u ) )
6		suceq 5790	. . . 4 |- ( v = u -> suc v = suc u )
7	6	fveq2d 6195	. . 3 |- ( v = u -> ( F ` suc v ) = ( F ` suc u ) )
8	5, 7	sseq12d 3634	. 2 |- ( v = u -> ( ( F ` v ) C_ ( F ` suc v ) <-> ( F ` u ) C_ ( F ` suc u ) ) )
9		fveq2 6191	. . 3 |- ( v = suc u -> ( F ` v ) = ( F ` suc u ) )
10		suceq 5790	. . . 4 |- ( v = suc u -> suc v = suc suc u )
11	10	fveq2d 6195	. . 3 |- ( v = suc u -> ( F ` suc v ) = ( F ` suc suc u ) )
12	9, 11	sseq12d 3634	. 2 |- ( v = suc u -> ( ( F ` v ) C_ ( F ` suc v ) <-> ( F ` suc u ) C_ ( F ` suc suc u ) ) )
13		fveq2 6191	. . 3 |- ( v = A -> ( F ` v ) = ( F ` A ) )
14		suceq 5790	. . . 4 |- ( v = A -> suc v = suc A )
15	14	fveq2d 6195	. . 3 |- ( v = A -> ( F ` suc v ) = ( F ` suc A ) )
16	13, 15	sseq12d 3634	. 2 |- ( v = A -> ( ( F ` v ) C_ ( F ` suc v ) <-> ( F ` A ) C_ ( F ` suc A ) ) )
17		inf3lem.1	. . . 4 |- G = ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } )
18		inf3lem.2	. . . 4 |- F = ( rec ( G , (/) ) |` om )
19		inf3lem.3	. . . 4 |- A e. _V
20	17, 18, 19, 19	inf3lemb 8522	. . 3 |- ( F ` (/) ) = (/)
21		0ss 3972	. . 3 |- (/) C_ ( F ` suc (/) )
22	20, 21	eqsstri 3635	. 2 |- ( F ` (/) ) C_ ( F ` suc (/) )
23		sstr2 3610	. . . . . . . 8 |- ( ( v i^i x ) C_ ( F ` u ) -> ( ( F ` u ) C_ ( F ` suc u ) -> ( v i^i x ) C_ ( F ` suc u ) ) )
24	23	com12 32	. . . . . . 7 |- ( ( F ` u ) C_ ( F ` suc u ) -> ( ( v i^i x ) C_ ( F ` u ) -> ( v i^i x ) C_ ( F ` suc u ) ) )
25	24	anim2d 589	. . . . . 6 |- ( ( F ` u ) C_ ( F ` suc u ) -> ( ( v e. x /\ ( v i^i x ) C_ ( F ` u ) ) -> ( v e. x /\ ( v i^i x ) C_ ( F ` suc u ) ) ) )
26		vex 3203	. . . . . . . . . 10 |- u e. _V
27	17, 18, 26, 19	inf3lemc 8523	. . . . . . . . 9 |- ( u e. om -> ( F ` suc u ) = ( G ` ( F ` u ) ) )
28	27	eleq2d 2687	. . . . . . . 8 |- ( u e. om -> ( v e. ( F ` suc u ) <-> v e. ( G ` ( F ` u ) ) ) )
29		vex 3203	. . . . . . . . 9 |- v e. _V
30		fvex 6201	. . . . . . . . 9 |- ( F ` u ) e. _V
31	17, 18, 29, 30	inf3lema 8521	. . . . . . . 8 |- ( v e. ( G ` ( F ` u ) ) <-> ( v e. x /\ ( v i^i x ) C_ ( F ` u ) ) )
32	28, 31	syl6bb 276	. . . . . . 7 |- ( u e. om -> ( v e. ( F ` suc u ) <-> ( v e. x /\ ( v i^i x ) C_ ( F ` u ) ) ) )
33		peano2b 7081	. . . . . . . . . 10 |- ( u e. om <-> suc u e. om )
34	26	sucex 7011	. . . . . . . . . . 11 |- suc u e. _V
35	17, 18, 34, 19	inf3lemc 8523	. . . . . . . . . 10 |- ( suc u e. om -> ( F ` suc suc u ) = ( G ` ( F ` suc u ) ) )
36	33, 35	sylbi 207	. . . . . . . . 9 |- ( u e. om -> ( F ` suc suc u ) = ( G ` ( F ` suc u ) ) )
37	36	eleq2d 2687	. . . . . . . 8 |- ( u e. om -> ( v e. ( F ` suc suc u ) <-> v e. ( G ` ( F ` suc u ) ) ) )
38		fvex 6201	. . . . . . . . 9 |- ( F ` suc u ) e. _V
39	17, 18, 29, 38	inf3lema 8521	. . . . . . . 8 |- ( v e. ( G ` ( F ` suc u ) ) <-> ( v e. x /\ ( v i^i x ) C_ ( F ` suc u ) ) )
40	37, 39	syl6bb 276	. . . . . . 7 |- ( u e. om -> ( v e. ( F ` suc suc u ) <-> ( v e. x /\ ( v i^i x ) C_ ( F ` suc u ) ) ) )
41	32, 40	imbi12d 334	. . . . . 6 |- ( u e. om -> ( ( v e. ( F ` suc u ) -> v e. ( F ` suc suc u ) ) <-> ( ( v e. x /\ ( v i^i x ) C_ ( F ` u ) ) -> ( v e. x /\ ( v i^i x ) C_ ( F ` suc u ) ) ) ) )
42	25, 41	syl5ibr 236	. . . . 5 |- ( u e. om -> ( ( F ` u ) C_ ( F ` suc u ) -> ( v e. ( F ` suc u ) -> v e. ( F ` suc suc u ) ) ) )
43	42	imp 445	. . . 4 |- ( ( u e. om /\ ( F ` u ) C_ ( F ` suc u ) ) -> ( v e. ( F ` suc u ) -> v e. ( F ` suc suc u ) ) )
44	43	ssrdv 3609	. . 3 |- ( ( u e. om /\ ( F ` u ) C_ ( F ` suc u ) ) -> ( F ` suc u ) C_ ( F ` suc suc u ) )
45	44	ex 450	. 2 |- ( u e. om -> ( ( F ` u ) C_ ( F ` suc u ) -> ( F ` suc u ) C_ ( F ` suc suc u ) ) )
46	4, 8, 12, 16, 22, 45	finds 7092	1 |- ( A e. om -> ( F ` A ) C_ ( F ` suc A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   |-> cmpt 4729   |` cres 5116   suc csuc 5725   ` cfv 5888   omcom 7065   reccrdg 7505

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-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-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-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-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506

This theorem is referenced by:  inf3lem4  8528