addge02d - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > addge02d		Unicode version

Theorem addge02d 8320

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 8259	. 2 $/\$
4	1, 2, 3	syl2anc 409	1

Colors of variables: wff set class

Syntax hints:

wi 4

wb 104

wcel 1481 class class class wbr 3937 (class class class)co 5782

cr 7643

cc0 7644

caddc 7647

cle 7825

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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-i2m1 7749 ax-0id 7752 ax-rnegex 7753 ax-pre-ltadd 7760

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-iota 5096 df-fv 5139 df-ov 5785 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830

This theorem is referenced by: nn2ge 8777 uzsubsubfz 9858 uzennn 10240 resqrexlemover 10814 resqrexlemdecn 10816 cvgratnnlemsumlt 11329 cosbnd 11496