Theorem nn1gt1 9176

Description: A positive integer is either one or greater than one. This is for NN; 0elnn 4717 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 2238	. . 3 |- ( x = 1 -> ( x = 1 <-> 1 = 1 ) )
2		breq2 4092	. . 3 |- ( x = 1 -> ( 1 < x <-> 1 < 1 ) )
3	1, 2	orbi12d 800	. 2 |- ( x = 1 -> ( ( x = 1 \/ 1 < x ) <-> ( 1 = 1 \/ 1 < 1 ) ) )
4		eqeq1 2238	. . 3 |- ( x = y -> ( x = 1 <-> y = 1 ) )
5		breq2 4092	. . 3 |- ( x = y -> ( 1 < x <-> 1 < y ) )
6	4, 5	orbi12d 800	. 2 |- ( x = y -> ( ( x = 1 \/ 1 < x ) <-> ( y = 1 \/ 1 < y ) ) )
7		eqeq1 2238	. . 3 |- ( x = ( y + 1 ) -> ( x = 1 <-> ( y + 1 ) = 1 ) )
8		breq2 4092	. . 3 |- ( x = ( y + 1 ) -> ( 1 < x <-> 1 < ( y + 1 ) ) )
9	7, 8	orbi12d 800	. 2 |- ( x = ( y + 1 ) -> ( ( x = 1 \/ 1 < x ) <-> ( ( y + 1 ) = 1 \/ 1 < ( y + 1 ) ) ) )
10		eqeq1 2238	. . 3 |- ( x = A -> ( x = 1 <-> A = 1 ) )
11		breq2 4092	. . 3 |- ( x = A -> ( 1 < x <-> 1 < A ) )
12	10, 11	orbi12d 800	. 2 |- ( x = A -> ( ( x = 1 \/ 1 < x ) <-> ( A = 1 \/ 1 < A ) ) )
13		eqid 2231	. . 3 |- 1 = 1
14	13	orci 738	. 2 |- ( 1 = 1 \/ 1 < 1 )
15		nngt0 9167	. . . . 5 |- ( y e. NN -> 0 < y )
16		nnre 9149	. . . . . 6 |- ( y e. NN -> y e. RR )
17		1re 8177	. . . . . 6 |- 1 e. RR
18		ltaddpos2 8632	. . . . . 6 |- ( ( y e. RR /\ 1 e. RR ) -> ( 0 < y <-> 1 < ( y + 1 ) ) )
19	16, 17, 18	sylancl 413	. . . . 5 |- ( y e. NN -> ( 0 < y <-> 1 < ( y + 1 ) ) )
20	15, 19	mpbid 147	. . . 4 |- ( y e. NN -> 1 < ( y + 1 ) )
21	20	olcd 741	. . 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 9158	1 |- ( A e. NN -> ( A = 1 \/ 1 < A ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 715   = wceq 1397   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   RRcr 8030   0cc0 8031   1c1 8032   + caddc 8034   < clt 8213   NNcn 9142

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-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  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-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-iota 5286  df-fv 5334  df-ov 6020  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-inn 9143

This theorem is referenced by:  nngt1ne1  9177  resqrexlemglsq  11582