Theorem inf3lem6 8530

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
inf3lem6	|- ( ( x =/= (/) /\ x C_ U. x ) -> F : om -1-1-> ~P x )

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 inf3lem6

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

Step	Hyp	Ref	Expression
1		inf3lem.1	. . . . . . . . . . 11 |- G = ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } )
2		inf3lem.2	. . . . . . . . . . 11 |- F = ( rec ( G , (/) ) |` om )
3		vex 3203	. . . . . . . . . . 11 |- u e. _V
4		vex 3203	. . . . . . . . . . 11 |- v e. _V
5	1, 2, 3, 4	inf3lem5 8529	. . . . . . . . . 10 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( u e. om /\ v e. u ) -> ( F ` v ) C. ( F ` u ) ) )
6		dfpss2 3692	. . . . . . . . . . 11 |- ( ( F ` v ) C. ( F ` u ) <-> ( ( F ` v ) C_ ( F ` u ) /\ -. ( F ` v ) = ( F ` u ) ) )
7	6	simprbi 480	. . . . . . . . . 10 |- ( ( F ` v ) C. ( F ` u ) -> -. ( F ` v ) = ( F ` u ) )
8	5, 7	syl6 35	. . . . . . . . 9 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( u e. om /\ v e. u ) -> -. ( F ` v ) = ( F ` u ) ) )
9	8	expdimp 453	. . . . . . . 8 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ u e. om ) -> ( v e. u -> -. ( F ` v ) = ( F ` u ) ) )
10	9	adantrl 752	. . . . . . 7 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. om /\ u e. om ) ) -> ( v e. u -> -. ( F ` v ) = ( F ` u ) ) )
11	1, 2, 4, 3	inf3lem5 8529	. . . . . . . . . 10 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( v e. om /\ u e. v ) -> ( F ` u ) C. ( F ` v ) ) )
12		dfpss2 3692	. . . . . . . . . . . 12 |- ( ( F ` u ) C. ( F ` v ) <-> ( ( F ` u ) C_ ( F ` v ) /\ -. ( F ` u ) = ( F ` v ) ) )
13	12	simprbi 480	. . . . . . . . . . 11 |- ( ( F ` u ) C. ( F ` v ) -> -. ( F ` u ) = ( F ` v ) )
14		eqcom 2629	. . . . . . . . . . 11 |- ( ( F ` u ) = ( F ` v ) <-> ( F ` v ) = ( F ` u ) )
15	13, 14	sylnib 318	. . . . . . . . . 10 |- ( ( F ` u ) C. ( F ` v ) -> -. ( F ` v ) = ( F ` u ) )
16	11, 15	syl6 35	. . . . . . . . 9 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( v e. om /\ u e. v ) -> -. ( F ` v ) = ( F ` u ) ) )
17	16	expdimp 453	. . . . . . . 8 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ v e. om ) -> ( u e. v -> -. ( F ` v ) = ( F ` u ) ) )
18	17	adantrr 753	. . . . . . 7 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. om /\ u e. om ) ) -> ( u e. v -> -. ( F ` v ) = ( F ` u ) ) )
19	10, 18	jaod 395	. . . . . 6 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. om /\ u e. om ) ) -> ( ( v e. u \/ u e. v ) -> -. ( F ` v ) = ( F ` u ) ) )
20	19	con2d 129	. . . . 5 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. om /\ u e. om ) ) -> ( ( F ` v ) = ( F ` u ) -> -. ( v e. u \/ u e. v ) ) )
21		nnord 7073	. . . . . . 7 |- ( v e. om -> Ord v )
22		nnord 7073	. . . . . . 7 |- ( u e. om -> Ord u )
23		ordtri3 5759	. . . . . . 7 |- ( ( Ord v /\ Ord u ) -> ( v = u <-> -. ( v e. u \/ u e. v ) ) )
24	21, 22, 23	syl2an 494	. . . . . 6 |- ( ( v e. om /\ u e. om ) -> ( v = u <-> -. ( v e. u \/ u e. v ) ) )
25	24	adantl 482	. . . . 5 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. om /\ u e. om ) ) -> ( v = u <-> -. ( v e. u \/ u e. v ) ) )
26	20, 25	sylibrd 249	. . . 4 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. om /\ u e. om ) ) -> ( ( F ` v ) = ( F ` u ) -> v = u ) )
27	26	ralrimivva 2971	. . 3 |- ( ( x =/= (/) /\ x C_ U. x ) -> A. v e. om A. u e. om ( ( F ` v ) = ( F ` u ) -> v = u ) )
28		frfnom 7530	. . . . . 6 |- ( rec ( G , (/) ) |` om ) Fn om
29		fneq1 5979	. . . . . 6 |- ( F = ( rec ( G , (/) ) |` om ) -> ( F Fn om <-> ( rec ( G , (/) ) |` om ) Fn om ) )
30	28, 29	mpbiri 248	. . . . 5 |- ( F = ( rec ( G , (/) ) |` om ) -> F Fn om )
31		fvelrnb 6243	. . . . . . . 8 |- ( F Fn om -> ( u e. ran F <-> E. v e. om ( F ` v ) = u ) )
32		inf3lem.4	. . . . . . . . . . . 12 |- B e. _V
33	1, 2, 4, 32	inf3lemd 8524	. . . . . . . . . . 11 |- ( v e. om -> ( F ` v ) C_ x )
34		fvex 6201	. . . . . . . . . . . 12 |- ( F ` v ) e. _V
35	34	elpw 4164	. . . . . . . . . . 11 |- ( ( F ` v ) e. ~P x <-> ( F ` v ) C_ x )
36	33, 35	sylibr 224	. . . . . . . . . 10 |- ( v e. om -> ( F ` v ) e. ~P x )
37		eleq1 2689	. . . . . . . . . 10 |- ( ( F ` v ) = u -> ( ( F ` v ) e. ~P x <-> u e. ~P x ) )
38	36, 37	syl5ibcom 235	. . . . . . . . 9 |- ( v e. om -> ( ( F ` v ) = u -> u e. ~P x ) )
39	38	rexlimiv 3027	. . . . . . . 8 |- ( E. v e. om ( F ` v ) = u -> u e. ~P x )
40	31, 39	syl6bi 243	. . . . . . 7 |- ( F Fn om -> ( u e. ran F -> u e. ~P x ) )
41	40	ssrdv 3609	. . . . . 6 |- ( F Fn om -> ran F C_ ~P x )
42	41	ancli 574	. . . . 5 |- ( F Fn om -> ( F Fn om /\ ran F C_ ~P x ) )
43	2, 30, 42	mp2b 10	. . . 4 |- ( F Fn om /\ ran F C_ ~P x )
44		df-f 5892	. . . 4 |- ( F : om --> ~P x <-> ( F Fn om /\ ran F C_ ~P x ) )
45	43, 44	mpbir 221	. . 3 |- F : om --> ~P x
46	27, 45	jctil 560	. 2 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( F : om --> ~P x /\ A. v e. om A. u e. om ( ( F ` v ) = ( F ` u ) -> v = u ) ) )
47		dff13 6512	. 2 |- ( F : om -1-1-> ~P x <-> ( F : om --> ~P x /\ A. v e. om A. u e. om ( ( F ` v ) = ( F ` u ) -> v = u ) ) )
48	46, 47	sylibr 224	1 |- ( ( x =/= (/) /\ x C_ U. x ) -> F : om -1-1-> ~P x )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   C. wpss 3575   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   |-> cmpt 4729   ran crn 5115   |` cres 5116   Ord word 5722   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   ` 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:  inf3lem7  8531  dominf  9267  dominfac  9395