addge02d - Intuitionistic Logic Explorer

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Theorem addge02d 8072

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 8012	. 2 $/\$
4	1, 2, 3	syl2anc 404	1

Colors of variables: wff set class

Syntax hints:

wi 4

wb 104

wcel 1439 class class class wbr 3851 (class class class)co 5666

cr 7410

cc0 7411

caddc 7414

cle 7584

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-addcom 7506 ax-addass 7508 ax-i2m1 7511 ax-0id 7514 ax-rnegex 7515 ax-pre-ltadd 7522

This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2622 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-xp 4458 df-cnv 4460 df-iota 4993 df-fv 5036 df-ov 5669 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589

This theorem is referenced by: nn2ge 8516 uzsubsubfz 9522 resqrexlemover 10504 resqrexlemdecn 10506 cvgratnnlemsumlt 10983 cosbnd 11105