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Metamath Proof Explorer

Theorem nlt1pi

Description: No positive integer is less than one. (Contributed by NM, 23-Mar-1996) (New usage is discouraged.)

Ref Expression
Assertion nlt1pi 
|- -. A

Proof

Step Hyp Ref Expression
1 elni 
 |-  ( A e. N. <-> ( A e. _om /\ A =/= (/) ) )
2 1 simprbi 
 |-  ( A e. N. -> A =/= (/) )
3 noel 
 |-  -. A e. (/)
4 1pi 
 |-  1o e. N.
5 ltpiord 
 |-  ( ( A e. N. /\ 1o e. N. ) -> ( A  A e. 1o ) )
6 4 5 mpan2 
 |-  ( A e. N. -> ( A  A e. 1o ) )
7 df-1o 
 |-  1o = suc (/)
8 7 eleq2i 
 |-  ( A e. 1o <-> A e. suc (/) )
9 elsucg 
 |-  ( A e. N. -> ( A e. suc (/) <-> ( A e. (/) \/ A = (/) ) ) )
10 8 9 bitrid 
 |-  ( A e. N. -> ( A e. 1o <-> ( A e. (/) \/ A = (/) ) ) )
11 6 10 bitrd 
 |-  ( A e. N. -> ( A  ( A e. (/) \/ A = (/) ) ) )
12 11 biimpa 
 |-  ( ( A e. N. /\ A  ( A e. (/) \/ A = (/) ) )
13 12 ord 
 |-  ( ( A e. N. /\ A  ( -. A e. (/) -> A = (/) ) )
14 3 13 mpi 
 |-  ( ( A e. N. /\ A  A = (/) )
15 14 ex 
 |-  ( A e. N. -> ( A  A = (/) ) )
16 15 necon3ad 
 |-  ( A e. N. -> ( A =/= (/) -> -. A
17 2 16 mpd 
 |-  ( A e. N. -> -. A
18 ltrelpi 
 |-
19 18 brel 
 |-  ( A  ( A e. N. /\ 1o e. N. ) )
20 19 simpld 
 |-  ( A  A e. N. )
21 20 con3i 
 |-  ( -. A e. N. -> -. A
22 17 21 pm2.61i 
 |-  -. A