Theorem ennnfonelemr 13292

Description: Lemma for ennnfone 13294. The interesting direction, expressed in deduction form. (Contributed by Jim Kingdon, 27-Oct-2022.)

Hypotheses

Ref	Expression
ennnfonelemr.dceq	|- ( ph -> A. x e. A A. y e. A DECID x = y )
ennnfonelemr.f	|- ( ph -> F : NN0 -onto-> A )
ennnfonelemr.n	|- ( ph -> A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) )

Assertion

Ref	Expression
ennnfonelemr	|- ( ph -> A ~~ NN )

Distinct variable groups:   y, A, x   n, F, j, k

Allowed substitution hints:   ph( x, y, j, k, n)   A( j, k, n)   F( x, y)

Proof of Theorem ennnfonelemr

Dummy variables a b d e f c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ennnfonelemr.dceq	. . 3 |- ( ph -> A. x e. A A. y e. A DECID x = y )
2		equequ1 1764	. . . . 5 |- ( x = a -> ( x = y <-> a = y ) )
3	2	dcbid 850	. . . 4 |- ( x = a -> (DECID x = y <-> DECID a = y ) )
4		equequ2 1765	. . . . 5 |- ( y = b -> ( a = y <-> a = b ) )
5	4	dcbid 850	. . . 4 |- ( y = b -> (DECID a = y <-> DECID a = b ) )
6	3, 5	cbvral2v 2799	. . 3 |- ( A. x e. A A. y e. A DECID x = y <-> A. a e. A A. b e. A DECID a = b )
7	1, 6	sylib 122	. 2 |- ( ph -> A. a e. A A. b e. A DECID a = b )
8		ennnfonelemr.f	. 2 |- ( ph -> F : NN0 -onto-> A )
9		ennnfonelemr.n	. . 3 |- ( ph -> A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) )
10		fveq2 5690	. . . . . . . . 9 |- ( j = f -> ( F ` j ) = ( F ` f ) )
11	10	neeq2d 2439	. . . . . . . 8 |- ( j = f -> ( ( F ` k ) =/= ( F ` j ) <-> ( F ` k ) =/= ( F ` f ) ) )
12	11	cbvralv 2786	. . . . . . 7 |- ( A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) <-> A. f e. ( 0 ... n ) ( F ` k ) =/= ( F ` f ) )
13	12	rexbii 2557	. . . . . 6 |- ( E. k e. NN0 A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) <-> E. k e. NN0 A. f e. ( 0 ... n ) ( F ` k ) =/= ( F ` f ) )
14		fveq2 5690	. . . . . . . . 9 |- ( k = e -> ( F ` k ) = ( F ` e ) )
15	14	neeq1d 2438	. . . . . . . 8 |- ( k = e -> ( ( F ` k ) =/= ( F ` f ) <-> ( F ` e ) =/= ( F ` f ) ) )
16	15	ralbidv 2550	. . . . . . 7 |- ( k = e -> ( A. f e. ( 0 ... n ) ( F ` k ) =/= ( F ` f ) <-> A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) ) )
17	16	cbvrexv 2787	. . . . . 6 |- ( E. k e. NN0 A. f e. ( 0 ... n ) ( F ` k ) =/= ( F ` f ) <-> E. e e. NN0 A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) )
18	13, 17	bitri 184	. . . . 5 |- ( E. k e. NN0 A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) <-> E. e e. NN0 A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) )
19	18	ralbii 2556	. . . 4 |- ( A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) <-> A. n e. NN0 E. e e. NN0 A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) )
20		oveq2 6083	. . . . . . 7 |- ( n = d -> ( 0 ... n ) = ( 0 ... d ) )
21	20	raleqdv 2755	. . . . . 6 |- ( n = d -> ( A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) <-> A. f e. ( 0 ... d ) ( F ` e ) =/= ( F ` f ) ) )
22	21	rexbidv 2551	. . . . 5 |- ( n = d -> ( E. e e. NN0 A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) <-> E. e e. NN0 A. f e. ( 0 ... d ) ( F ` e ) =/= ( F ` f ) ) )
23	22	cbvralv 2786	. . . 4 |- ( A. n e. NN0 E. e e. NN0 A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) <-> A. d e. NN0 E. e e. NN0 A. f e. ( 0 ... d ) ( F ` e ) =/= ( F ` f ) )
24	19, 23	bitri 184	. . 3 |- ( A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) <-> A. d e. NN0 E. e e. NN0 A. f e. ( 0 ... d ) ( F ` e ) =/= ( F ` f ) )
25	9, 24	sylib 122	. 2 |- ( ph -> A. d e. NN0 E. e e. NN0 A. f e. ( 0 ... d ) ( F ` e ) =/= ( F ` f ) )
26		oveq1 6082	. . . 4 |- ( c = a -> ( c + 1 ) = ( a + 1 ) )
27	26	cbvmptv 4222	. . 3 |- ( c e. ZZ |-> ( c + 1 ) ) = ( a e. ZZ |-> ( a + 1 ) )
28		freceq1 6653	. . 3 |- ( ( c e. ZZ |-> ( c + 1 ) ) = ( a e. ZZ |-> ( a + 1 ) ) -> frec ( ( c e. ZZ |-> ( c + 1 ) ) , 0 ) = frec ( ( a e. ZZ |-> ( a + 1 ) ) , 0 ) )
29	27, 28	ax-mp 5	. 2 |- frec ( ( c e. ZZ |-> ( c + 1 ) ) , 0 ) = frec ( ( a e. ZZ |-> ( a + 1 ) ) , 0 )
30	7, 8, 25, 29	ennnfonelemnn0 13291	1 |- ( ph -> A ~~ NN )

Syntax hints:   -> wi 4  DECID wdc 846   = wceq 1402   =/= wne 2420   A.wral 2528   E.wrex 2529   class class class wbr 4125   |-> cmpt 4187   -onto->wfo 5370   ` cfv 5372  (class class class)co 6075  freccfrec 6651   ~~ cen 7010   0cc0 8169   1c1 8170   + caddc 8172   NNcn 9283   NN0cn0 9542   ZZcz 9623   ...cfz 10390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-er 6797  df-pm 6915  df-en 7013  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-seqfrec 10863

This theorem is referenced by:  ennnfone  13294