nn0p1gt0 - Intuitionistic Logic Explorer

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Theorem nn0p1gt0 9295

Description: A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.)

**Assertion**
Ref	Expression
nn0p1gt0

**Proof of Theorem nn0p1gt0**
Step	Hyp	Ref	Expression
1		nn0re 9275	. 2
2		1red 8058	. 2
3		nn0ge0 9291	. 2
4		0lt1 8170	. . 3
5	4	a1i 9	. 2
6	1, 2, 3, 5	addgegt0d 8563	1

Colors of variables: wff set class

Syntax hints:

wi 4

wcel 2167 class class class wbr 4034 (class class class)co 5925

cc0 7896

c1 7897

caddc 7899

clt 8078

cn0 9266

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-addcom 7996 ax-addass 7998 ax-i2m1 8001 ax-0lt1 8002 ax-0id 8004 ax-rnegex 8005 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-ltadd 8012

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-rab 2484 df-v 2765 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-br 4035 df-opab 4096 df-xp 4670 df-cnv 4672 df-iota 5220 df-fv 5267 df-ov 5928 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-inn 9008 df-n0 9267

This theorem is referenced by: ubmelm1fzo 10319 nn0ltexp2 10818 ennnfonelemp1 12648