Theorem peano5nnnn 8111

Description: Peano's inductive postulate. This is a counterpart to peano5nni 9145 designed for real number axioms which involve natural numbers (notably, axcaucvg 8119). (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 6024	. . . 4 |- ( y = z -> ( y + 1 ) = ( z + 1 ) )
2	1	eleq1d 2300	. . 3 |- ( y = z -> ( ( y + 1 ) e. A <-> ( z + 1 ) e. A ) )
3	2	cbvralv 2767	. 2 |- ( A. y e. A ( y + 1 ) e. A <-> A. z e. A ( z + 1 ) e. A )
4		ax1re 8081	. . . . 5 |- 1 e. RR
5		elin 3390	. . . . . 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 3427	. . . . . 6 |- ( A i^i RR ) C_ A
9		ssralv 3291	. . . . . 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 3428	. . . . . . . 8 |- ( A i^i RR ) C_ RR
12	11	sseli 3223	. . . . . . 7 |- ( y e. ( A i^i RR ) -> y e. RR )
13		axaddrcl 8084	. . . . . . . 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 3390	. . . . . . . 8 |- ( ( y + 1 ) e. ( A i^i RR ) <-> ( ( y + 1 ) e. A /\ ( y + 1 ) e. RR ) )
16	15	simplbi2com 1489	. . . . . . 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 2593	. . . . 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 8078	. . . . . . 7 |- CC e. _V
21		axresscn 8079	. . . . . . 7 |- RR C_ CC
22	20, 21	ssexi 4227	. . . . . 6 |- RR e. _V
23	22	inex2 4224	. . . . 5 |- ( A i^i RR ) e. _V
24		eleq2 2295	. . . . . . . 8 |- ( x = ( A i^i RR ) -> ( 1 e. x <-> 1 e. ( A i^i RR ) ) )
25		eleq2 2295	. . . . . . . . 9 |- ( x = ( A i^i RR ) -> ( ( y + 1 ) e. x <-> ( y + 1 ) e. ( A i^i RR ) ) )
26	25	raleqbi1dv 2742	. . . . . . . 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 2952	. . . . . 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 3943	. . . . . . 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 3273	. . . . . 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 3239	. 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 1397   e. wcel 2202   {cab 2217   A.wral 2510   _Vcvv 2802   i^i cin 3199   C_ wss 3200   |^|cint 3928  (class class class)co 6017   CCcc 8029   RRcr 8030   1c1 8032   + caddc 8034

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-2o 6582  df-oadd 6585  df-omul 6586  df-er 6701  df-ec 6703  df-qs 6707  df-ni 7523  df-pli 7524  df-mi 7525  df-lti 7526  df-plpq 7563  df-mpq 7564  df-enq 7566  df-nqqs 7567  df-plqqs 7568  df-mqqs 7569  df-1nqqs 7570  df-rq 7571  df-ltnqqs 7572  df-enq0 7643  df-nq0 7644  df-0nq0 7645  df-plq0 7646  df-mq0 7647  df-inp 7685  df-i1p 7686  df-iplp 7687  df-enr 7945  df-nr 7946  df-plr 7947  df-0r 7950  df-1r 7951  df-c 8037  df-1 8039  df-r 8041  df-add 8042

This theorem is referenced by:  nnindnn  8112