Theorem peano5nnnn 7893

Description: Peano's inductive postulate. This is a counterpart to peano5nni 8924 designed for real number axioms which involve natural numbers (notably, axcaucvg 7901). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nntopi.n	|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }

Assertion

Ref	Expression
peano5nnnn	|- ( ( 1 e. A /\ A. z e. A ( z + 1 ) e. A ) -> N C_ A )

Distinct variable groups:   x, y, A   z, A, y

Allowed substitution hints:   N( x, y, z)

Proof of Theorem peano5nnnn

Step	Hyp	Ref	Expression
1		oveq1 5884	. . . 4 |- ( y = z -> ( y + 1 ) = ( z + 1 ) )
2	1	eleq1d 2246	. . 3 |- ( y = z -> ( ( y + 1 ) e. A <-> ( z + 1 ) e. A ) )
3	2	cbvralv 2705	. 2 |- ( A. y e. A ( y + 1 ) e. A <-> A. z e. A ( z + 1 ) e. A )
4		ax1re 7863	. . . . 5 |- 1 e. RR
5		elin 3320	. . . . . 6 |- ( 1 e. ( A i^i RR ) <-> ( 1 e. A /\ 1 e. RR ) )
6	5	biimpri 133	. . . . 5 |- ( ( 1 e. A /\ 1 e. RR ) -> 1 e. ( A i^i RR ) )
7	4, 6	mpan2 425	. . . 4 |- ( 1 e. A -> 1 e. ( A i^i RR ) )
8		inss1 3357	. . . . . 6 |- ( A i^i RR ) C_ A
9		ssralv 3221	. . . . . 6 |- ( ( A i^i RR ) C_ A -> ( A. y e. A ( y + 1 ) e. A -> A. y e. ( A i^i RR ) ( y + 1 ) e. A ) )
10	8, 9	ax-mp 5	. . . . 5 |- ( A. y e. A ( y + 1 ) e. A -> A. y e. ( A i^i RR ) ( y + 1 ) e. A )
11		inss2 3358	. . . . . . . 8 |- ( A i^i RR ) C_ RR
12	11	sseli 3153	. . . . . . 7 |- ( y e. ( A i^i RR ) -> y e. RR )
13		axaddrcl 7866	. . . . . . . 8 |- ( ( y e. RR /\ 1 e. RR ) -> ( y + 1 ) e. RR )
14	4, 13	mpan2 425	. . . . . . 7 |- ( y e. RR -> ( y + 1 ) e. RR )
15		elin 3320	. . . . . . . 8 |- ( ( y + 1 ) e. ( A i^i RR ) <-> ( ( y + 1 ) e. A /\ ( y + 1 ) e. RR ) )
16	15	simplbi2com 1444	. . . . . . 7 |- ( ( y + 1 ) e. RR -> ( ( y + 1 ) e. A -> ( y + 1 ) e. ( A i^i RR ) ) )
17	12, 14, 16	3syl 17	. . . . . 6 |- ( y e. ( A i^i RR ) -> ( ( y + 1 ) e. A -> ( y + 1 ) e. ( A i^i RR ) ) )
18	17	ralimia 2538	. . . . 5 |- ( A. y e. ( A i^i RR ) ( y + 1 ) e. A -> A. y e. ( A i^i RR ) ( y + 1 ) e. ( A i^i RR ) )
19	10, 18	syl 14	. . . 4 |- ( A. y e. A ( y + 1 ) e. A -> A. y e. ( A i^i RR ) ( y + 1 ) e. ( A i^i RR ) )
20		axcnex 7860	. . . . . . 7 |- CC e. _V
21		axresscn 7861	. . . . . . 7 |- RR C_ CC
22	20, 21	ssexi 4143	. . . . . 6 |- RR e. _V
23	22	inex2 4140	. . . . 5 |- ( A i^i RR ) e. _V
24		eleq2 2241	. . . . . . . 8 |- ( x = ( A i^i RR ) -> ( 1 e. x <-> 1 e. ( A i^i RR ) ) )
25		eleq2 2241	. . . . . . . . 9 |- ( x = ( A i^i RR ) -> ( ( y + 1 ) e. x <-> ( y + 1 ) e. ( A i^i RR ) ) )
26	25	raleqbi1dv 2681	. . . . . . . 8 |- ( x = ( A i^i RR ) -> ( A. y e. x ( y + 1 ) e. x <-> A. y e. ( A i^i RR ) ( y + 1 ) e. ( A i^i RR ) ) )
27	24, 26	anbi12d 473	. . . . . . 7 |- ( x = ( A i^i RR ) -> ( ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) <-> ( 1 e. ( A i^i RR ) /\ A. y e. ( A i^i RR ) ( y + 1 ) e. ( A i^i RR ) ) ) )
28	27	elabg 2885	. . . . . 6 |- ( ( A i^i RR ) e. _V -> ( ( A i^i RR ) e. { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } <-> ( 1 e. ( A i^i RR ) /\ A. y e. ( A i^i RR ) ( y + 1 ) e. ( A i^i RR ) ) ) )
29		nntopi.n	. . . . . . 7 |- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }
30		intss1 3861	. . . . . . 7 |- ( ( A i^i RR ) e. { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } -> |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } C_ ( A i^i RR ) )
31	29, 30	eqsstrid 3203	. . . . . 6 |- ( ( A i^i RR ) e. { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } -> N C_ ( A i^i RR ) )
32	28, 31	syl6bir 164	. . . . 5 |- ( ( A i^i RR ) e. _V -> ( ( 1 e. ( A i^i RR ) /\ A. y e. ( A i^i RR ) ( y + 1 ) e. ( A i^i RR ) ) -> N C_ ( A i^i RR ) ) )
33	23, 32	ax-mp 5	. . . 4 |- ( ( 1 e. ( A i^i RR ) /\ A. y e. ( A i^i RR ) ( y + 1 ) e. ( A i^i RR ) ) -> N C_ ( A i^i RR ) )
34	7, 19, 33	syl2an 289	. . 3 |- ( ( 1 e. A /\ A. y e. A ( y + 1 ) e. A ) -> N C_ ( A i^i RR ) )
35	34, 8	sstrdi 3169	. 2 |- ( ( 1 e. A /\ A. y e. A ( y + 1 ) e. A ) -> N C_ A )
36	3, 35	sylan2br 288	1 |- ( ( 1 e. A /\ A. z e. A ( z + 1 ) e. A ) -> N C_ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   {cab 2163   A.wral 2455   _Vcvv 2739   i^i cin 3130   C_ wss 3131   |^|cint 3846  (class class class)co 5877   CCcc 7811   RRcr 7812   1c1 7814   + caddc 7816

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-2o 6420  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-enq0 7425  df-nq0 7426  df-0nq0 7427  df-plq0 7428  df-mq0 7429  df-inp 7467  df-i1p 7468  df-iplp 7469  df-enr 7727  df-nr 7728  df-plr 7729  df-0r 7732  df-1r 7733  df-c 7819  df-1 7821  df-r 7823  df-add 7824

This theorem is referenced by:  nnindnn  7894