Theorem nn0ge2m1nnALT 10018

Description: Alternate proof of nn0ge2m1nn 9627: If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. This version is proved using eluz2 9927, a theorem for upper sets of integers, which are defined later than the positive and nonnegative integers. This proof is, however, much shorter than the proof of nn0ge2m1nn 9627. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
nn0ge2m1nnALT	|- ( ( N e. NN0 /\ 2 <_ N ) -> ( N - 1 ) e. NN )

Proof of Theorem nn0ge2m1nnALT

Step	Hyp	Ref	Expression
1		2z 9672	. . . 4 |- 2 e. ZZ
2	1	a1i 9	. . 3 |- ( ( N e. NN0 /\ 2 <_ N ) -> 2 e. ZZ )
3		nn0z 9664	. . . 4 |- ( N e. NN0 -> N e. ZZ )
4	3	adantr 276	. . 3 |- ( ( N e. NN0 /\ 2 <_ N ) -> N e. ZZ )
5		simpr 110	. . 3 |- ( ( N e. NN0 /\ 2 <_ N ) -> 2 <_ N )
6		eluz2 9927	. . 3 |- ( N e. ( ZZ>= ` 2 ) <-> ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) )
7	2, 4, 5, 6	syl3anbrc 1212	. 2 |- ( ( N e. NN0 /\ 2 <_ N ) -> N e. ( ZZ>= ` 2 ) )
8		uz2m1nn 10005	. 2 |- ( N e. ( ZZ>= ` 2 ) -> ( N - 1 ) e. NN )
9	7, 8	syl 14	1 |- ( ( N e. NN0 /\ 2 <_ N ) -> ( N - 1 ) e. NN )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   1c1 8180   <_ cle 8361   - cmin 8497   NNcn 9304   2c2 9355   NN0cn0 9563   ZZcz 9644   ZZ>=cuz 9921

This proof depends on 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-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  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-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-uz 9922

This theorem is used by: (None)