Theorem inf3lem6 8039

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8041 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 3109	. . . . . . . . . . 11 |- u e. _V
4		vex 3109	. . . . . . . . . . 11 |- v e. _V
5	1, 2, 3, 4	inf3lem5 8038	. . . . . . . . . 10 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( u e. om /\ v e. u ) -> ( F ` v ) C. ( F ` u ) ) )
6		dfpss2 3582	. . . . . . . . . . 11 |- ( ( F ` v ) C. ( F ` u ) <-> ( ( F ` v ) C_ ( F ` u ) /\ -. ( F ` v ) = ( F ` u ) ) )
7	6	simprbi 464	. . . . . . . . . 10 |- ( ( F ` v ) C. ( F ` u ) -> -. ( F ` v ) = ( F ` u ) )
8	5, 7	syl6 33	. . . . . . . . 9 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( u e. om /\ v e. u ) -> -. ( F ` v ) = ( F ` u ) ) )
9	8	expdimp 437	. . . . . . . 8 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ u e. om ) -> ( v e. u -> -. ( F ` v ) = ( F ` u ) ) )
10	9	adantrl 715	. . . . . . 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 8038	. . . . . . . . . 10 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( v e. om /\ u e. v ) -> ( F ` u ) C. ( F ` v ) ) )
12		dfpss2 3582	. . . . . . . . . . . 12 |- ( ( F ` u ) C. ( F ` v ) <-> ( ( F ` u ) C_ ( F ` v ) /\ -. ( F ` u ) = ( F ` v ) ) )
13	12	simprbi 464	. . . . . . . . . . 11 |- ( ( F ` u ) C. ( F ` v ) -> -. ( F ` u ) = ( F ` v ) )
14		eqcom 2469	. . . . . . . . . . 11 |- ( ( F ` u ) = ( F ` v ) <-> ( F ` v ) = ( F ` u ) )
15	13, 14	sylnib 304	. . . . . . . . . 10 |- ( ( F ` u ) C. ( F ` v ) -> -. ( F ` v ) = ( F ` u ) )
16	11, 15	syl6 33	. . . . . . . . 9 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( v e. om /\ u e. v ) -> -. ( F ` v ) = ( F ` u ) ) )
17	16	expdimp 437	. . . . . . . 8 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ v e. om ) -> ( u e. v -> -. ( F ` v ) = ( F ` u ) ) )
18	17	adantrr 716	. . . . . . 7 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. om /\ u e. om ) ) -> ( u e. v -> -. ( F ` v ) = ( F ` u ) ) )
19	10, 18	jaod 380	. . . . . 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 115	. . . . 5 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. om /\ u e. om ) ) -> ( ( F ` v ) = ( F ` u ) -> -. ( v e. u \/ u e. v ) ) )
21		nnord 6679	. . . . . . 7 |- ( v e. om -> Ord v )
22		nnord 6679	. . . . . . 7 |- ( u e. om -> Ord u )
23		ordtri3 4907	. . . . . . 7 |- ( ( Ord v /\ Ord u ) -> ( v = u <-> -. ( v e. u \/ u e. v ) ) )
24	21, 22, 23	syl2an 477	. . . . . 6 |- ( ( v e. om /\ u e. om ) -> ( v = u <-> -. ( v e. u \/ u e. v ) ) )
25	24	adantl 466	. . . . 5 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. om /\ u e. om ) ) -> ( v = u <-> -. ( v e. u \/ u e. v ) ) )
26	20, 25	sylibrd 234	. . . 4 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. om /\ u e. om ) ) -> ( ( F ` v ) = ( F ` u ) -> v = u ) )
27	26	ralrimivva 2878	. . 3 |- ( ( x =/= (/) /\ x C_ U. x ) -> A. v e. om A. u e. om ( ( F ` v ) = ( F ` u ) -> v = u ) )
28		frfnom 7090	. . . . . 6 |- ( rec ( G , (/) ) |` om ) Fn om
29		fneq1 5660	. . . . . 6 |- ( F = ( rec ( G , (/) ) |` om ) -> ( F Fn om <-> ( rec ( G , (/) ) |` om ) Fn om ) )
30	28, 29	mpbiri 233	. . . . 5 |- ( F = ( rec ( G , (/) ) |` om ) -> F Fn om )
31		fvelrnb 5906	. . . . . . . 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 8033	. . . . . . . . . . 11 |- ( v e. om -> ( F ` v ) C_ x )
34		fvex 5867	. . . . . . . . . . . 12 |- ( F ` v ) e. _V
35	34	elpw 4009	. . . . . . . . . . 11 |- ( ( F ` v ) e. ~P x <-> ( F ` v ) C_ x )
36	33, 35	sylibr 212	. . . . . . . . . 10 |- ( v e. om -> ( F ` v ) e. ~P x )
37		eleq1 2532	. . . . . . . . . 10 |- ( ( F ` v ) = u -> ( ( F ` v ) e. ~P x <-> u e. ~P x ) )
38	36, 37	syl5ibcom 220	. . . . . . . . 9 |- ( v e. om -> ( ( F ` v ) = u -> u e. ~P x ) )
39	38	rexlimiv 2942	. . . . . . . 8 |- ( E. v e. om ( F ` v ) = u -> u e. ~P x )
40	31, 39	syl6bi 228	. . . . . . 7 |- ( F Fn om -> ( u e. ran F -> u e. ~P x ) )
41	40	ssrdv 3503	. . . . . 6 |- ( F Fn om -> ran F C_ ~P x )
42	41	ancli 551	. . . . 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 5583	. . . 4 |- ( F : om --> ~P x <-> ( F Fn om /\ ran F C_ ~P x ) )
45	43, 44	mpbir 209	. . 3 |- F : om --> ~P x
46	27, 45	jctil 537	. 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 6145	. 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 212	1 |- ( ( x =/= (/) /\ x C_ U. x ) -> F : om -1-1-> ~P x )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1374   e. wcel 1762   =/= wne 2655   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3106   i^i cin 3468   C_ wss 3469   C. wpss 3470   (/)c0 3778   ~Pcpw 4003   U.cuni 4238   |-> cmpt 4498   Ord word 4870   ran crn 4993   |` cres 4994   Fn wfn 5574   -->wf 5575   -1-1->wf1 5576   ` cfv 5579   omcom 6671   reccrdg 7065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-reg 8007

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-om 6672  df-recs 7032  df-rdg 7066

This theorem is referenced by:  inf3lem7  8040  dominf  8814  dominfac  8937