Theorem nnltp1c 6263

Description: Any natural is less than one plus itself. (Contributed by SF, 25-Mar-2015.)

Assertion

Ref	Expression
nnltp1c	|- (N e. Nn -> N <c (N +o 1c))

Proof of Theorem nnltp1c

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

Step	Hyp	Ref	Expression
1		nnltp1clem1 6262	. 2 |- {x | x <c (x +o 1c)} e. _V
2		id 19	. . 3 |- (x = 0c -> x = 0c)
3		addceq1 4384	. . 3 |- (x = 0c -> (x +o 1c) = (0c +o 1c))
4	2, 3	breq12d 4653	. 2 |- (x = 0c -> (x <c (x +o 1c) <-> 0c <c (0c +o 1c)))
5		id 19	. . 3 |- (x = n -> x = n)
6		addceq1 4384	. . 3 |- (x = n -> (x +o 1c) = (n +o 1c))
7	5, 6	breq12d 4653	. 2 |- (x = n -> (x <c (x +o 1c) <-> n <c (n +o 1c)))
8		id 19	. . 3 |- (x = (n +o 1c) -> x = (n +o 1c))
9		addceq1 4384	. . 3 |- (x = (n +o 1c) -> (x +o 1c) = ((n +o 1c) +o 1c))
10	8, 9	breq12d 4653	. 2 |- (x = (n +o 1c) -> (x <c (x +o 1c) <-> (n +o 1c) <c ((n +o 1c) +o 1c)))
11		id 19	. . 3 |- (x = N -> x = N)
12		addceq1 4384	. . 3 |- (x = N -> (x +o 1c) = (N +o 1c))
13	11, 12	breq12d 4653	. 2 |- (x = N -> (x <c (x +o 1c) <-> N <c (N +o 1c)))
14		0cnc 6139	. . . 4 |- 0c e. NC
15		1cnc 6140	. . . 4 |- 1c e. NC
16		addlecncs 6210	. . . 4 |- ((0c e. NC /\ 1c e. NC ) -> 0c <_c (0c +o 1c))
17	14, 15, 16	mp2an 653	. . 3 |- 0c <_c (0c +o 1c)
18		0cnsuc 4402	. . . 4 |- (0c +o 1c) =/= 0c
19	18	necomi 2599	. . 3 |- 0c =/= (0c +o 1c)
20		brltc 6115	. . 3 |- (0c <c (0c +o 1c) <-> (0c <_c (0c +o 1c) /\ 0c =/= (0c +o 1c)))
21	17, 19, 20	mpbir2an 886	. 2 |- 0c <c (0c +o 1c)
22		nnnc 6147	. . . . 5 |- (n e. Nn -> n e. NC )
23		peano2nc 6146	. . . . . 6 |- (n e. NC -> (n +o 1c) e. NC )
24	22, 23	syl 15	. . . . 5 |- (n e. Nn -> (n +o 1c) e. NC )
25		leaddc1 6215	. . . . . . 7 |- (((n e. NC /\ (n +o 1c) e. NC /\ 1c e. NC ) /\ n <_c (n +o 1c)) -> (n +o 1c) <_c ((n +o 1c) +o 1c))
26	25	ex 423	. . . . . 6 |- ((n e. NC /\ (n +o 1c) e. NC /\ 1c e. NC ) -> (n <_c (n +o 1c) -> (n +o 1c) <_c ((n +o 1c) +o 1c)))
27	15, 26	mp3an3 1266	. . . . 5 |- ((n e. NC /\ (n +o 1c) e. NC ) -> (n <_c (n +o 1c) -> (n +o 1c) <_c ((n +o 1c) +o 1c)))
28	22, 24, 27	syl2anc 642	. . . 4 |- (n e. Nn -> (n <_c (n +o 1c) -> (n +o 1c) <_c ((n +o 1c) +o 1c)))
29		peano2 4404	. . . . . . 7 |- (n e. Nn -> (n +o 1c) e. Nn )
30		suc11nnc 4559	. . . . . . 7 |- ((n e. Nn /\ (n +o 1c) e. Nn ) -> ((n +o 1c) = ((n +o 1c) +o 1c) <-> n = (n +o 1c)))
31	29, 30	mpdan 649	. . . . . 6 |- (n e. Nn -> ((n +o 1c) = ((n +o 1c) +o 1c) <-> n = (n +o 1c)))
32	31	biimpd 198	. . . . 5 |- (n e. Nn -> ((n +o 1c) = ((n +o 1c) +o 1c) -> n = (n +o 1c)))
33	32	necon3d 2555	. . . 4 |- (n e. Nn -> (n =/= (n +o 1c) -> (n +o 1c) =/= ((n +o 1c) +o 1c)))
34	28, 33	anim12d 546	. . 3 |- (n e. Nn -> ((n <_c (n +o 1c) /\ n =/= (n +o 1c)) -> ((n +o 1c) <_c ((n +o 1c) +o 1c) /\ (n +o 1c) =/= ((n +o 1c) +o 1c))))
35		brltc 6115	. . 3 |- (n <c (n +o 1c) <-> (n <_c (n +o 1c) /\ n =/= (n +o 1c)))
36		brltc 6115	. . 3 |- ((n +o 1c) <c ((n +o 1c) +o 1c) <-> ((n +o 1c) <_c ((n +o 1c) +o 1c) /\ (n +o 1c) =/= ((n +o 1c) +o 1c)))
37	34, 35, 36	3imtr4g 261	. 2 |- (n e. Nn -> (n <c (n +o 1c) -> (n +o 1c) <c ((n +o 1c) +o 1c)))
38	1, 4, 7, 10, 13, 21, 37	finds 4412	1 |- (N e. Nn -> N <c (N +o 1c))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710   =/= wne 2517  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +o cplc 4376   class class class wbr 4640   NC cncs 6089   <__c clec 6090   <_c cltc 6091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-fix 5741  df-cup 5743  df-disj 5745  df-addcfn 5747  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-lec 6100  df-ltc 6101  df-nc 6102

This theorem is referenced by:  nmembers1lem3  6271