addge02d - Intuitionistic Logic Explorer

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Theorem addge02d 8289

Description: A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)

**Hypotheses**
Ref	Expression
leidd.1
ltnegd.2

**Assertion**
Ref	Expression
addge02d

**Proof of Theorem addge02d**
Step	Hyp	Ref	Expression
1		leidd.1	. 2
2		ltnegd.2	. 2
3		addge02 8228	. 2 $/\$
4	1, 2, 3	syl2anc 408	1

Colors of variables: wff set class

Syntax hints:

wi 4

wb 104

wcel 1480 class class class wbr 3924 (class class class)co 5767

cr 7612

cc0 7613

caddc 7616

cle 7794

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-pre-ltadd 7729

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-cnv 4542 df-iota 5083 df-fv 5126 df-ov 5770 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799

This theorem is referenced by: nn2ge 8746 uzsubsubfz 9820 uzennn 10202 resqrexlemover 10775 resqrexlemdecn 10777 cvgratnnlemsumlt 11290 cosbnd 11449