Theorem inf3lem5 8529

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8532 for detailed description. (Contributed by NM, 29-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
inf3lem5	|- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( A e. om /\ B e. A ) -> ( F ` B ) C. ( F ` 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 inf3lem5

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

Step	Hyp	Ref	Expression
1		elnn 7075	. . . 4 |- ( ( B e. A /\ A e. om ) -> B e. om )
2	1	ancoms 469	. . 3 |- ( ( A e. om /\ B e. A ) -> B e. om )
3		nnord 7073	. . . . . . 7 |- ( A e. om -> Ord A )
4		ordsucss 7018	. . . . . . 7 |- ( Ord A -> ( B e. A -> suc B C_ A ) )
5	3, 4	syl 17	. . . . . 6 |- ( A e. om -> ( B e. A -> suc B C_ A ) )
6	5	adantr 481	. . . . 5 |- ( ( A e. om /\ B e. om ) -> ( B e. A -> suc B C_ A ) )
7		peano2b 7081	. . . . . 6 |- ( B e. om <-> suc B e. om )
8		fveq2 6191	. . . . . . . . . 10 |- ( v = suc B -> ( F ` v ) = ( F ` suc B ) )
9	8	psseq2d 3700	. . . . . . . . 9 |- ( v = suc B -> ( ( F ` B ) C. ( F ` v ) <-> ( F ` B ) C. ( F ` suc B ) ) )
10	9	imbi2d 330	. . . . . . . 8 |- ( v = suc B -> ( ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` v ) ) <-> ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` suc B ) ) ) )
11		fveq2 6191	. . . . . . . . . 10 |- ( v = u -> ( F ` v ) = ( F ` u ) )
12	11	psseq2d 3700	. . . . . . . . 9 |- ( v = u -> ( ( F ` B ) C. ( F ` v ) <-> ( F ` B ) C. ( F ` u ) ) )
13	12	imbi2d 330	. . . . . . . 8 |- ( v = u -> ( ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` v ) ) <-> ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` u ) ) ) )
14		fveq2 6191	. . . . . . . . . 10 |- ( v = suc u -> ( F ` v ) = ( F ` suc u ) )
15	14	psseq2d 3700	. . . . . . . . 9 |- ( v = suc u -> ( ( F ` B ) C. ( F ` v ) <-> ( F ` B ) C. ( F ` suc u ) ) )
16	15	imbi2d 330	. . . . . . . 8 |- ( v = suc u -> ( ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` v ) ) <-> ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` suc u ) ) ) )
17		fveq2 6191	. . . . . . . . . 10 |- ( v = A -> ( F ` v ) = ( F ` A ) )
18	17	psseq2d 3700	. . . . . . . . 9 |- ( v = A -> ( ( F ` B ) C. ( F ` v ) <-> ( F ` B ) C. ( F ` A ) ) )
19	18	imbi2d 330	. . . . . . . 8 |- ( v = A -> ( ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` v ) ) <-> ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` A ) ) ) )
20		inf3lem.1	. . . . . . . . . . 11 |- G = ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } )
21		inf3lem.2	. . . . . . . . . . 11 |- F = ( rec ( G , (/) ) |` om )
22		inf3lem.4	. . . . . . . . . . 11 |- B e. _V
23	20, 21, 22, 22	inf3lem4 8528	. . . . . . . . . 10 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( B e. om -> ( F ` B ) C. ( F ` suc B ) ) )
24	23	com12 32	. . . . . . . . 9 |- ( B e. om -> ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` suc B ) ) )
25	7, 24	sylbir 225	. . . . . . . 8 |- ( suc B e. om -> ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` suc B ) ) )
26		vex 3203	. . . . . . . . . . . 12 |- u e. _V
27	20, 21, 26, 22	inf3lem4 8528	. . . . . . . . . . 11 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( u e. om -> ( F ` u ) C. ( F ` suc u ) ) )
28		psstr 3711	. . . . . . . . . . . 12 |- ( ( ( F ` B ) C. ( F ` u ) /\ ( F ` u ) C. ( F ` suc u ) ) -> ( F ` B ) C. ( F ` suc u ) )
29	28	expcom 451	. . . . . . . . . . 11 |- ( ( F ` u ) C. ( F ` suc u ) -> ( ( F ` B ) C. ( F ` u ) -> ( F ` B ) C. ( F ` suc u ) ) )
30	27, 29	syl6com 37	. . . . . . . . . 10 |- ( u e. om -> ( ( x =/= (/) /\ x C_ U. x ) -> ( ( F ` B ) C. ( F ` u ) -> ( F ` B ) C. ( F ` suc u ) ) ) )
31	30	a2d 29	. . . . . . . . 9 |- ( u e. om -> ( ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` u ) ) -> ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` suc u ) ) ) )
32	31	ad2antrr 762	. . . . . . . 8 |- ( ( ( u e. om /\ suc B e. om ) /\ suc B C_ u ) -> ( ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` u ) ) -> ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` suc u ) ) ) )
33	10, 13, 16, 19, 25, 32	findsg 7093	. . . . . . 7 |- ( ( ( A e. om /\ suc B e. om ) /\ suc B C_ A ) -> ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` A ) ) )
34	33	ex 450	. . . . . 6 |- ( ( A e. om /\ suc B e. om ) -> ( suc B C_ A -> ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` A ) ) ) )
35	7, 34	sylan2b 492	. . . . 5 |- ( ( A e. om /\ B e. om ) -> ( suc B C_ A -> ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` A ) ) ) )
36	6, 35	syld 47	. . . 4 |- ( ( A e. om /\ B e. om ) -> ( B e. A -> ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` A ) ) ) )
37	36	impancom 456	. . 3 |- ( ( A e. om /\ B e. A ) -> ( B e. om -> ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` A ) ) ) )
38	2, 37	mpd 15	. 2 |- ( ( A e. om /\ B e. A ) -> ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` B ) C. ( F ` A ) ) )
39	38	com12 32	1 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( A e. om /\ B e. A ) -> ( F ` B ) C. ( F ` A ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   C. wpss 3575   (/)c0 3915   U.cuni 4436   |-> cmpt 4729   |` cres 5116   Ord word 5722   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  ax-reg 8497

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:  inf3lem6  8530