Theorem nn1gt1 8901

Description: A positive integer is either one or greater than one. This is for NN; 0elnn 4601 is a similar theorem for om (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.)

Assertion

Ref	Expression
nn1gt1	|- ( A e. NN -> ( A = 1 \/ 1 < A ) )

Proof of Theorem nn1gt1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2177	. . 3 |- ( x = 1 -> ( x = 1 <-> 1 = 1 ) )
2		breq2 3991	. . 3 |- ( x = 1 -> ( 1 < x <-> 1 < 1 ) )
3	1, 2	orbi12d 788	. 2 |- ( x = 1 -> ( ( x = 1 \/ 1 < x ) <-> ( 1 = 1 \/ 1 < 1 ) ) )
4		eqeq1 2177	. . 3 |- ( x = y -> ( x = 1 <-> y = 1 ) )
5		breq2 3991	. . 3 |- ( x = y -> ( 1 < x <-> 1 < y ) )
6	4, 5	orbi12d 788	. 2 |- ( x = y -> ( ( x = 1 \/ 1 < x ) <-> ( y = 1 \/ 1 < y ) ) )
7		eqeq1 2177	. . 3 |- ( x = ( y + 1 ) -> ( x = 1 <-> ( y + 1 ) = 1 ) )
8		breq2 3991	. . 3 |- ( x = ( y + 1 ) -> ( 1 < x <-> 1 < ( y + 1 ) ) )
9	7, 8	orbi12d 788	. 2 |- ( x = ( y + 1 ) -> ( ( x = 1 \/ 1 < x ) <-> ( ( y + 1 ) = 1 \/ 1 < ( y + 1 ) ) ) )
10		eqeq1 2177	. . 3 |- ( x = A -> ( x = 1 <-> A = 1 ) )
11		breq2 3991	. . 3 |- ( x = A -> ( 1 < x <-> 1 < A ) )
12	10, 11	orbi12d 788	. 2 |- ( x = A -> ( ( x = 1 \/ 1 < x ) <-> ( A = 1 \/ 1 < A ) ) )
13		eqid 2170	. . 3 |- 1 = 1
14	13	orci 726	. 2 |- ( 1 = 1 \/ 1 < 1 )
15		nngt0 8892	. . . . 5 |- ( y e. NN -> 0 < y )
16		nnre 8874	. . . . . 6 |- ( y e. NN -> y e. RR )
17		1re 7908	. . . . . 6 |- 1 e. RR
18		ltaddpos2 8361	. . . . . 6 |- ( ( y e. RR /\ 1 e. RR ) -> ( 0 < y <-> 1 < ( y + 1 ) ) )
19	16, 17, 18	sylancl 411	. . . . 5 |- ( y e. NN -> ( 0 < y <-> 1 < ( y + 1 ) ) )
20	15, 19	mpbid 146	. . . 4 |- ( y e. NN -> 1 < ( y + 1 ) )
21	20	olcd 729	. . 3 |- ( y e. NN -> ( ( y + 1 ) = 1 \/ 1 < ( y + 1 ) ) )
22	21	a1d 22	. 2 |- ( y e. NN -> ( ( y = 1 \/ 1 < y ) -> ( ( y + 1 ) = 1 \/ 1 < ( y + 1 ) ) ) )
23	3, 6, 9, 12, 14, 22	nnind 8883	1 |- ( A e. NN -> ( A = 1 \/ 1 < A ) )

Syntax hints:   -> wi 4   <-> wb 104   \/ wo 703   = wceq 1348   e. wcel 2141   class class class wbr 3987  (class class class)co 5851   RRcr 7762   0cc0 7763   1c1 7764   + caddc 7766   < clt 7943   NNcn 8867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7854  ax-resscn 7855  ax-1cn 7856  ax-1re 7857  ax-icn 7858  ax-addcl 7859  ax-addrcl 7860  ax-mulcl 7861  ax-addcom 7863  ax-addass 7865  ax-i2m1 7868  ax-0lt1 7869  ax-0id 7871  ax-rnegex 7872  ax-pre-ltirr 7875  ax-pre-ltwlin 7876  ax-pre-lttrn 7877  ax-pre-ltadd 7879

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-br 3988  df-opab 4049  df-xp 4615  df-cnv 4617  df-iota 5158  df-fv 5204  df-ov 5854  df-pnf 7945  df-mnf 7946  df-xr 7947  df-ltxr 7948  df-le 7949  df-inn 8868

This theorem is referenced by:  nngt1ne1  8902  resqrexlemglsq  10975