Theorem peano5nnnn 7976

Description: Peano's inductive postulate. This is a counterpart to peano5nni 9010 designed for real number axioms which involve natural numbers (notably, axcaucvg 7984). (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 5932	. . . 4 |- ( y = z -> ( y + 1 ) = ( z + 1 ) )
2	1	eleq1d 2265	. . 3 |- ( y = z -> ( ( y + 1 ) e. A <-> ( z + 1 ) e. A ) )
3	2	cbvralv 2729	. 2 |- ( A. y e. A ( y + 1 ) e. A <-> A. z e. A ( z + 1 ) e. A )
4		ax1re 7946	. . . . 5 |- 1 e. RR
5		elin 3347	. . . . . 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 3384	. . . . . 6 |- ( A i^i RR ) C_ A
9		ssralv 3248	. . . . . 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 3385	. . . . . . . 8 |- ( A i^i RR ) C_ RR
12	11	sseli 3180	. . . . . . 7 |- ( y e. ( A i^i RR ) -> y e. RR )
13		axaddrcl 7949	. . . . . . . 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 3347	. . . . . . . 8 |- ( ( y + 1 ) e. ( A i^i RR ) <-> ( ( y + 1 ) e. A /\ ( y + 1 ) e. RR ) )
16	15	simplbi2com 1455	. . . . . . 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 2558	. . . . 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 7943	. . . . . . 7 |- CC e. _V
21		axresscn 7944	. . . . . . 7 |- RR C_ CC
22	20, 21	ssexi 4172	. . . . . 6 |- RR e. _V
23	22	inex2 4169	. . . . 5 |- ( A i^i RR ) e. _V
24		eleq2 2260	. . . . . . . 8 |- ( x = ( A i^i RR ) -> ( 1 e. x <-> 1 e. ( A i^i RR ) ) )
25		eleq2 2260	. . . . . . . . 9 |- ( x = ( A i^i RR ) -> ( ( y + 1 ) e. x <-> ( y + 1 ) e. ( A i^i RR ) ) )
26	25	raleqbi1dv 2705	. . . . . . . 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 2910	. . . . . 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 3890	. . . . . . 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 3230	. . . . . 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	biimtrrdi 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 3196	. 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 1364   e. wcel 2167   {cab 2182   A.wral 2475   _Vcvv 2763   i^i cin 3156   C_ wss 3157   |^|cint 3875  (class class class)co 5925   CCcc 7894   RRcr 7895   1c1 7897   + caddc 7899

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-i1p 7551  df-iplp 7552  df-enr 7810  df-nr 7811  df-plr 7812  df-0r 7815  df-1r 7816  df-c 7902  df-1 7904  df-r 7906  df-add 7907

This theorem is referenced by:  nnindnn  7977