Theorem nlt1pi 9728

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

Assertion

Ref	Expression
nlt1pi	|- -. A <N 1o

Proof of Theorem nlt1pi

Step	Hyp	Ref	Expression
1		elni 9698	. . . 4 |- ( A e. N. <-> ( A e. om /\ A =/= (/) ) )
2	1	simprbi 480	. . 3 |- ( A e. N. -> A =/= (/) )
3		noel 3919	. . . . . 6 |- -. A e. (/)
4		1pi 9705	. . . . . . . . . 10 |- 1o e. N.
5		ltpiord 9709	. . . . . . . . . 10 |- ( ( A e. N. /\ 1o e. N. ) -> ( A <N 1o <-> A e. 1o ) )
6	4, 5	mpan2 707	. . . . . . . . 9 |- ( A e. N. -> ( A <N 1o <-> A e. 1o ) )
7		df-1o 7560	. . . . . . . . . . 11 |- 1o = suc (/)
8	7	eleq2i 2693	. . . . . . . . . 10 |- ( A e. 1o <-> A e. suc (/) )
9		elsucg 5792	. . . . . . . . . 10 |- ( A e. N. -> ( A e. suc (/) <-> ( A e. (/) \/ A = (/) ) ) )
10	8, 9	syl5bb 272	. . . . . . . . 9 |- ( A e. N. -> ( A e. 1o <-> ( A e. (/) \/ A = (/) ) ) )
11	6, 10	bitrd 268	. . . . . . . 8 |- ( A e. N. -> ( A <N 1o <-> ( A e. (/) \/ A = (/) ) ) )
12	11	biimpa 501	. . . . . . 7 |- ( ( A e. N. /\ A <N 1o ) -> ( A e. (/) \/ A = (/) ) )
13	12	ord 392	. . . . . 6 |- ( ( A e. N. /\ A <N 1o ) -> ( -. A e. (/) -> A = (/) ) )
14	3, 13	mpi 20	. . . . 5 |- ( ( A e. N. /\ A <N 1o ) -> A = (/) )
15	14	ex 450	. . . 4 |- ( A e. N. -> ( A <N 1o -> A = (/) ) )
16	15	necon3ad 2807	. . 3 |- ( A e. N. -> ( A =/= (/) -> -. A <N 1o ) )
17	2, 16	mpd 15	. 2 |- ( A e. N. -> -. A <N 1o )
18		ltrelpi 9711	. . . . 5 |- <N C_ ( N. X. N. )
19	18	brel 5168	. . . 4 |- ( A <N 1o -> ( A e. N. /\ 1o e. N. ) )
20	19	simpld 475	. . 3 |- ( A <N 1o -> A e. N. )
21	20	con3i 150	. 2 |- ( -. A e. N. -> -. A <N 1o )
22	17, 21	pm2.61i 176	1 |- -. A <N 1o

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   class class class wbr 4653   suc csuc 5725   omcom 7065   1oc1o 7553   N.cnpi 9666   <N clti 9669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-om 7066  df-1o 7560  df-ni 9694  df-lti 9697

This theorem is referenced by:  indpi  9729  pinq  9749  archnq  9802