Theorem inf3lem6 8135

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8137 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 3047	. . . . . . . . . . 11 |- u e. _V
4		vex 3047	. . . . . . . . . . 11 |- v e. _V
5	1, 2, 3, 4	inf3lem5 8134	. . . . . . . . . 10 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( u e. om /\ v e. u ) -> ( F ` v ) C. ( F ` u ) ) )
6		dfpss2 3517	. . . . . . . . . . 11 |- ( ( F ` v ) C. ( F ` u ) <-> ( ( F ` v ) C_ ( F ` u ) /\ -. ( F ` v ) = ( F ` u ) ) )
7	6	simprbi 466	. . . . . . . . . 10 |- ( ( F ` v ) C. ( F ` u ) -> -. ( F ` v ) = ( F ` u ) )
8	5, 7	syl6 34	. . . . . . . . 9 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( u e. om /\ v e. u ) -> -. ( F ` v ) = ( F ` u ) ) )
9	8	expdimp 439	. . . . . . . 8 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ u e. om ) -> ( v e. u -> -. ( F ` v ) = ( F ` u ) ) )
10	9	adantrl 721	. . . . . . 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 8134	. . . . . . . . . 10 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( v e. om /\ u e. v ) -> ( F ` u ) C. ( F ` v ) ) )
12		dfpss2 3517	. . . . . . . . . . . 12 |- ( ( F ` u ) C. ( F ` v ) <-> ( ( F ` u ) C_ ( F ` v ) /\ -. ( F ` u ) = ( F ` v ) ) )
13	12	simprbi 466	. . . . . . . . . . 11 |- ( ( F ` u ) C. ( F ` v ) -> -. ( F ` u ) = ( F ` v ) )
14		eqcom 2457	. . . . . . . . . . 11 |- ( ( F ` u ) = ( F ` v ) <-> ( F ` v ) = ( F ` u ) )
15	13, 14	sylnib 306	. . . . . . . . . 10 |- ( ( F ` u ) C. ( F ` v ) -> -. ( F ` v ) = ( F ` u ) )
16	11, 15	syl6 34	. . . . . . . . 9 |- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( v e. om /\ u e. v ) -> -. ( F ` v ) = ( F ` u ) ) )
17	16	expdimp 439	. . . . . . . 8 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ v e. om ) -> ( u e. v -> -. ( F ` v ) = ( F ` u ) ) )
18	17	adantrr 722	. . . . . . 7 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. om /\ u e. om ) ) -> ( u e. v -> -. ( F ` v ) = ( F ` u ) ) )
19	10, 18	jaod 382	. . . . . 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 119	. . . . 5 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. om /\ u e. om ) ) -> ( ( F ` v ) = ( F ` u ) -> -. ( v e. u \/ u e. v ) ) )
21		nnord 6697	. . . . . . 7 |- ( v e. om -> Ord v )
22		nnord 6697	. . . . . . 7 |- ( u e. om -> Ord u )
23		ordtri3 5458	. . . . . . 7 |- ( ( Ord v /\ Ord u ) -> ( v = u <-> -. ( v e. u \/ u e. v ) ) )
24	21, 22, 23	syl2an 480	. . . . . 6 |- ( ( v e. om /\ u e. om ) -> ( v = u <-> -. ( v e. u \/ u e. v ) ) )
25	24	adantl 468	. . . . 5 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. om /\ u e. om ) ) -> ( v = u <-> -. ( v e. u \/ u e. v ) ) )
26	20, 25	sylibrd 238	. . . 4 |- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. om /\ u e. om ) ) -> ( ( F ` v ) = ( F ` u ) -> v = u ) )
27	26	ralrimivva 2808	. . 3 |- ( ( x =/= (/) /\ x C_ U. x ) -> A. v e. om A. u e. om ( ( F ` v ) = ( F ` u ) -> v = u ) )
28		frfnom 7149	. . . . . 6 |- ( rec ( G , (/) ) |` om ) Fn om
29		fneq1 5662	. . . . . 6 |- ( F = ( rec ( G , (/) ) |` om ) -> ( F Fn om <-> ( rec ( G , (/) ) |` om ) Fn om ) )
30	28, 29	mpbiri 237	. . . . 5 |- ( F = ( rec ( G , (/) ) |` om ) -> F Fn om )
31		fvelrnb 5910	. . . . . . . 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 8129	. . . . . . . . . . 11 |- ( v e. om -> ( F ` v ) C_ x )
34		fvex 5873	. . . . . . . . . . . 12 |- ( F ` v ) e. _V
35	34	elpw 3956	. . . . . . . . . . 11 |- ( ( F ` v ) e. ~P x <-> ( F ` v ) C_ x )
36	33, 35	sylibr 216	. . . . . . . . . 10 |- ( v e. om -> ( F ` v ) e. ~P x )
37		eleq1 2516	. . . . . . . . . 10 |- ( ( F ` v ) = u -> ( ( F ` v ) e. ~P x <-> u e. ~P x ) )
38	36, 37	syl5ibcom 224	. . . . . . . . 9 |- ( v e. om -> ( ( F ` v ) = u -> u e. ~P x ) )
39	38	rexlimiv 2872	. . . . . . . 8 |- ( E. v e. om ( F ` v ) = u -> u e. ~P x )
40	31, 39	syl6bi 232	. . . . . . 7 |- ( F Fn om -> ( u e. ran F -> u e. ~P x ) )
41	40	ssrdv 3437	. . . . . 6 |- ( F Fn om -> ran F C_ ~P x )
42	41	ancli 554	. . . . 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 5585	. . . 4 |- ( F : om --> ~P x <-> ( F Fn om /\ ran F C_ ~P x ) )
45	43, 44	mpbir 213	. . 3 |- F : om --> ~P x
46	27, 45	jctil 540	. 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 6157	. 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 216	1 |- ( ( x =/= (/) /\ x C_ U. x ) -> F : om -1-1-> ~P x )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   = wceq 1443   e. wcel 1886   =/= wne 2621   A.wral 2736   E.wrex 2737   {crab 2740   _Vcvv 3044   i^i cin 3402   C_ wss 3403   C. wpss 3404   (/)c0 3730   ~Pcpw 3950   U.cuni 4197   |-> cmpt 4460   ran crn 4834   |` cres 4835   Ord word 5421   Fn wfn 5576   -->wf 5577   -1-1->wf1 5578   ` cfv 5581   omcom 6689   reccrdg 7124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-reg 8104

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6690  df-wrecs 7025  df-recs 7087  df-rdg 7125

This theorem is referenced by:  inf3lem7  8136  dominf  8872  dominfac  8995