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Mirrors > Home > ILE Home > Th. List > 1lt2pi | Unicode version |
Description: One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) |
Ref | Expression |
---|---|
1lt2pi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 6424 | . . . . 5 | |
2 | nna0 6378 | . . . . 5 | |
3 | 1, 2 | ax-mp 5 | . . . 4 |
4 | 0lt1o 6345 | . . . . 5 | |
5 | peano1 4516 | . . . . . 6 | |
6 | nnaord 6413 | . . . . . 6 | |
7 | 5, 1, 1, 6 | mp3an 1316 | . . . . 5 |
8 | 4, 7 | mpbi 144 | . . . 4 |
9 | 3, 8 | eqeltrri 2214 | . . 3 |
10 | 1pi 7147 | . . . 4 | |
11 | addpiord 7148 | . . . 4 | |
12 | 10, 10, 11 | mp2an 423 | . . 3 |
13 | 9, 12 | eleqtrri 2216 | . 2 |
14 | addclpi 7159 | . . . 4 | |
15 | 10, 10, 14 | mp2an 423 | . . 3 |
16 | ltpiord 7151 | . . 3 | |
17 | 10, 15, 16 | mp2an 423 | . 2 |
18 | 13, 17 | mpbir 145 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 104 wceq 1332 wcel 1481 c0 3368 class class class wbr 3937 com 4512 (class class class)co 5782 c1o 6314 coa 6318 cnpi 7104 cpli 7105 clti 7107 |
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-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 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-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-oadd 6325 df-ni 7136 df-pli 7137 df-lti 7139 |
This theorem is referenced by: 1lt2nq 7238 |
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