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Mirrors > Home > ILE Home > Th. List > nn2ge | Unicode version |
Description: There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.) |
Ref | Expression |
---|---|
nn2ge |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnaddcl 8708 | . 2 | |
2 | 0red 7735 | . . . 4 | |
3 | nnre 8695 | . . . . 5 | |
4 | 3 | adantl 275 | . . . 4 |
5 | nngt0 8713 | . . . . 5 | |
6 | 5 | adantl 275 | . . . 4 |
7 | 2, 4, 6 | ltled 7849 | . . 3 |
8 | nnre 8695 | . . . . 5 | |
9 | 8 | adantr 274 | . . . 4 |
10 | 9, 4 | addge01d 8263 | . . 3 |
11 | 7, 10 | mpbid 146 | . 2 |
12 | nngt0 8713 | . . . . 5 | |
13 | 12 | adantr 274 | . . . 4 |
14 | 2, 9, 13 | ltled 7849 | . . 3 |
15 | 4, 9 | addge02d 8264 | . . 3 |
16 | 14, 15 | mpbid 146 | . 2 |
17 | breq2 3903 | . . . 4 | |
18 | breq2 3903 | . . . 4 | |
19 | 17, 18 | anbi12d 464 | . . 3 |
20 | 19 | rspcev 2763 | . 2 |
21 | 1, 11, 16, 20 | syl12anc 1199 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 wrex 2394 class class class wbr 3899 (class class class)co 5742 cr 7587 cc0 7588 caddc 7591 clt 7768 cle 7769 cn 8688 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-xp 4515 df-cnv 4517 df-iota 5058 df-fv 5101 df-ov 5745 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-inn 8689 |
This theorem is referenced by: (None) |
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