Theorem inf3lem3 8527

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8532 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 8500. (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
inf3lem3	|- ( ( x =/= (/) /\ x C_ U. x ) -> ( A e. om -> ( F ` A ) =/= ( 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 inf3lem3

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inf3lem.1	. . . 4 |- G = ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } )
2		inf3lem.2	. . . 4 |- F = ( rec ( G , (/) ) |` om )
3		inf3lem.3	. . . 4 |- A e. _V
4		inf3lem.4	. . . 4 |- B e. _V
5	1, 2, 3, 4	inf3lemd 8524	. . 3 |- ( A e. om -> ( F ` A ) C_ x )
6	1, 2, 3, 4	inf3lem2 8526	. . . 4 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( A e. om -> ( F ` A ) =/= x ) )
7	6	com12 32	. . 3 |- ( A e. om -> ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` A ) =/= x ) )
8		pssdifn0 3944	. . 3 |- ( ( ( F ` A ) C_ x /\ ( F ` A ) =/= x ) -> ( x \ ( F ` A ) ) =/= (/) )
9	5, 7, 8	syl6an 568	. 2 |- ( A e. om -> ( ( x =/= (/) /\ x C_ U. x ) -> ( x \ ( F ` A ) ) =/= (/) ) )
10		vex 3203	. . . . 5 |- x e. _V
11	10	difexi 4809	. . . 4 |- ( x \ ( F ` A ) ) e. _V
12		zfreg 8500	. . . 4 |- ( ( ( x \ ( F ` A ) ) e. _V /\ ( x \ ( F ` A ) ) =/= (/) ) -> E. v e. ( x \ ( F ` A ) ) ( v i^i ( x \ ( F ` A ) ) ) = (/) )
13	11, 12	mpan 706	. . 3 |- ( ( x \ ( F ` A ) ) =/= (/) -> E. v e. ( x \ ( F ` A ) ) ( v i^i ( x \ ( F ` A ) ) ) = (/) )
14		eldifi 3732	. . . . . . . . . 10 |- ( v e. ( x \ ( F ` A ) ) -> v e. x )
15		inssdif0 3947	. . . . . . . . . . 11 |- ( ( v i^i x ) C_ ( F ` A ) <-> ( v i^i ( x \ ( F ` A ) ) ) = (/) )
16	15	biimpri 218	. . . . . . . . . 10 |- ( ( v i^i ( x \ ( F ` A ) ) ) = (/) -> ( v i^i x ) C_ ( F ` A ) )
17	14, 16	anim12i 590	. . . . . . . . 9 |- ( ( v e. ( x \ ( F ` A ) ) /\ ( v i^i ( x \ ( F ` A ) ) ) = (/) ) -> ( v e. x /\ ( v i^i x ) C_ ( F ` A ) ) )
18		vex 3203	. . . . . . . . . 10 |- v e. _V
19		fvex 6201	. . . . . . . . . 10 |- ( F ` A ) e. _V
20	1, 2, 18, 19	inf3lema 8521	. . . . . . . . 9 |- ( v e. ( G ` ( F ` A ) ) <-> ( v e. x /\ ( v i^i x ) C_ ( F ` A ) ) )
21	17, 20	sylibr 224	. . . . . . . 8 |- ( ( v e. ( x \ ( F ` A ) ) /\ ( v i^i ( x \ ( F ` A ) ) ) = (/) ) -> v e. ( G ` ( F ` A ) ) )
22	1, 2, 3, 4	inf3lemc 8523	. . . . . . . . 9 |- ( A e. om -> ( F ` suc A ) = ( G ` ( F ` A ) ) )
23	22	eleq2d 2687	. . . . . . . 8 |- ( A e. om -> ( v e. ( F ` suc A ) <-> v e. ( G ` ( F ` A ) ) ) )
24	21, 23	syl5ibr 236	. . . . . . 7 |- ( A e. om -> ( ( v e. ( x \ ( F ` A ) ) /\ ( v i^i ( x \ ( F ` A ) ) ) = (/) ) -> v e. ( F ` suc A ) ) )
25		eldifn 3733	. . . . . . . . 9 |- ( v e. ( x \ ( F ` A ) ) -> -. v e. ( F ` A ) )
26	25	adantr 481	. . . . . . . 8 |- ( ( v e. ( x \ ( F ` A ) ) /\ ( v i^i ( x \ ( F ` A ) ) ) = (/) ) -> -. v e. ( F ` A ) )
27	26	a1i 11	. . . . . . 7 |- ( A e. om -> ( ( v e. ( x \ ( F ` A ) ) /\ ( v i^i ( x \ ( F ` A ) ) ) = (/) ) -> -. v e. ( F ` A ) ) )
28	24, 27	jcad 555	. . . . . 6 |- ( A e. om -> ( ( v e. ( x \ ( F ` A ) ) /\ ( v i^i ( x \ ( F ` A ) ) ) = (/) ) -> ( v e. ( F ` suc A ) /\ -. v e. ( F ` A ) ) ) )
29		eleq2 2690	. . . . . . . . 9 |- ( ( F ` A ) = ( F ` suc A ) -> ( v e. ( F ` A ) <-> v e. ( F ` suc A ) ) )
30	29	biimprd 238	. . . . . . . 8 |- ( ( F ` A ) = ( F ` suc A ) -> ( v e. ( F ` suc A ) -> v e. ( F ` A ) ) )
31		iman 440	. . . . . . . 8 |- ( ( v e. ( F ` suc A ) -> v e. ( F ` A ) ) <-> -. ( v e. ( F ` suc A ) /\ -. v e. ( F ` A ) ) )
32	30, 31	sylib 208	. . . . . . 7 |- ( ( F ` A ) = ( F ` suc A ) -> -. ( v e. ( F ` suc A ) /\ -. v e. ( F ` A ) ) )
33	32	necon2ai 2823	. . . . . 6 |- ( ( v e. ( F ` suc A ) /\ -. v e. ( F ` A ) ) -> ( F ` A ) =/= ( F ` suc A ) )
34	28, 33	syl6 35	. . . . 5 |- ( A e. om -> ( ( v e. ( x \ ( F ` A ) ) /\ ( v i^i ( x \ ( F ` A ) ) ) = (/) ) -> ( F ` A ) =/= ( F ` suc A ) ) )
35	34	expd 452	. . . 4 |- ( A e. om -> ( v e. ( x \ ( F ` A ) ) -> ( ( v i^i ( x \ ( F ` A ) ) ) = (/) -> ( F ` A ) =/= ( F ` suc A ) ) ) )
36	35	rexlimdv 3030	. . 3 |- ( A e. om -> ( E. v e. ( x \ ( F ` A ) ) ( v i^i ( x \ ( F ` A ) ) ) = (/) -> ( F ` A ) =/= ( F ` suc A ) ) )
37	13, 36	syl5 34	. 2 |- ( A e. om -> ( ( x \ ( F ` A ) ) =/= (/) -> ( F ` A ) =/= ( F ` suc A ) ) )
38	9, 37	syldc 48	1 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( A e. om -> ( F ` A ) =/= ( F ` suc A ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   U.cuni 4436   |-> 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  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:  inf3lem4  8528