Theorem inf3lem6 8085

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8087 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 3064	. . . . . . . . . . 11 |- u e. _V
4		vex 3064	. . . . . . . . . . 11 |- v e. _V
5	1, 2, 3, 4	inf3lem5 8084	. . . . . . . . . 10 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( u e. om /\ v e. u ) -> ( F ` v ) C. ( F ` u ) ) )
6		dfpss2 3530	. . . . . . . . . . 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 716	. . . . . . 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 8084	. . . . . . . . . 10 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( v e. om /\ u e. v ) -> ( F ` u ) C. ( F ` v ) ) )
12		dfpss2 3530	. . . . . . . . . . . 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 2413	. . . . . . . . . . 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 717	. . . . . . 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 117	. . . . 5 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. om /\ u e. om ) ) -> ( ( F ` v ) = ( F ` u ) -> -. ( v e. u \/ u e. v ) ) )
21		nnord 6693	. . . . . . 7 |- ( v e. om -> Ord v )
22		nnord 6693	. . . . . . 7 |- ( u e. om -> Ord u )
23		ordtri3 5448	. . . . . . 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 236	. . . 4 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. om /\ u e. om ) ) -> ( ( F ` v ) = ( F ` u ) -> v = u ) )
27	26	ralrimivva 2827	. . 3 |- ( ( x =/= (/) /\ x C_ U. x ) -> A. v e. om A. u e. om ( ( F ` v ) = ( F ` u ) -> v = u ) )
28		frfnom 7139	. . . . . 6 |- ( rec ( G , (/) ) |` om ) Fn om
29		fneq1 5652	. . . . . 6 |- ( F = ( rec ( G , (/) ) |` om ) -> ( F Fn om <-> ( rec ( G , (/) ) |` om ) Fn om ) )
30	28, 29	mpbiri 235	. . . . 5 |- ( F = ( rec ( G , (/) ) |` om ) -> F Fn om )
31		fvelrnb 5898	. . . . . . . 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 8079	. . . . . . . . . . 11 |- ( v e. om -> ( F ` v ) C_ x )
34		fvex 5861	. . . . . . . . . . . 12 |- ( F ` v ) e. _V
35	34	elpw 3963	. . . . . . . . . . 11 |- ( ( F ` v ) e. ~P x <-> ( F ` v ) C_ x )
36	33, 35	sylibr 214	. . . . . . . . . 10 |- ( v e. om -> ( F ` v ) e. ~P x )
37		eleq1 2476	. . . . . . . . . 10 |- ( ( F ` v ) = u -> ( ( F ` v ) e. ~P x <-> u e. ~P x ) )
38	36, 37	syl5ibcom 222	. . . . . . . . 9 |- ( v e. om -> ( ( F ` v ) = u -> u e. ~P x ) )
39	38	rexlimiv 2892	. . . . . . . 8 |- ( E. v e. om ( F ` v ) = u -> u e. ~P x )
40	31, 39	syl6bi 230	. . . . . . 7 |- ( F Fn om -> ( u e. ran F -> u e. ~P x ) )
41	40	ssrdv 3450	. . . . . 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 5575	. . . 4 |- ( F : om --> ~P x <-> ( F Fn om /\ ran F C_ ~P x ) )
45	43, 44	mpbir 211	. . 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 6149	. 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 214	1 |- ( ( x =/= (/) /\ x C_ U. x ) -> F : om -1-1-> ~P x )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 186   \/ wo 368   /\ wa 369   = wceq 1407   e. wcel 1844   =/= wne 2600   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3061   i^i cin 3415   C_ wss 3416   C. wpss 3417   (/)c0 3740   ~Pcpw 3957   U.cuni 4193   |-> cmpt 4455   ran crn 4826   |` cres 4827   Ord word 5411   Fn wfn 5566   -->wf 5567   -1-1->wf1 5568   ` cfv 5571   omcom 6685   reccrdg 7114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-reg 8054

This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-om 6686  df-wrecs 7015  df-recs 7077  df-rdg 7115

This theorem is referenced by:  inf3lem7  8086  dominf  8859  dominfac  8982