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Mirrors > Home > ILE Home > Th. List > 1lt2pi | GIF version |
Description: One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) |
Ref | Expression |
---|---|
1lt2pi | ⊢ 1𝑜 <N (1𝑜 +N 1𝑜) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 6181 | . . . . 5 ⊢ 1𝑜 ∈ ω | |
2 | nna0 6139 | . . . . 5 ⊢ (1𝑜 ∈ ω → (1𝑜 +𝑜 ∅) = 1𝑜) | |
3 | 1, 2 | ax-mp 7 | . . . 4 ⊢ (1𝑜 +𝑜 ∅) = 1𝑜 |
4 | 0lt1o 6108 | . . . . 5 ⊢ ∅ ∈ 1𝑜 | |
5 | peano1 4364 | . . . . . 6 ⊢ ∅ ∈ ω | |
6 | nnaord 6170 | . . . . . 6 ⊢ ((∅ ∈ ω ∧ 1𝑜 ∈ ω ∧ 1𝑜 ∈ ω) → (∅ ∈ 1𝑜 ↔ (1𝑜 +𝑜 ∅) ∈ (1𝑜 +𝑜 1𝑜))) | |
7 | 5, 1, 1, 6 | mp3an 1269 | . . . . 5 ⊢ (∅ ∈ 1𝑜 ↔ (1𝑜 +𝑜 ∅) ∈ (1𝑜 +𝑜 1𝑜)) |
8 | 4, 7 | mpbi 143 | . . . 4 ⊢ (1𝑜 +𝑜 ∅) ∈ (1𝑜 +𝑜 1𝑜) |
9 | 3, 8 | eqeltrri 2156 | . . 3 ⊢ 1𝑜 ∈ (1𝑜 +𝑜 1𝑜) |
10 | 1pi 6603 | . . . 4 ⊢ 1𝑜 ∈ N | |
11 | addpiord 6604 | . . . 4 ⊢ ((1𝑜 ∈ N ∧ 1𝑜 ∈ N) → (1𝑜 +N 1𝑜) = (1𝑜 +𝑜 1𝑜)) | |
12 | 10, 10, 11 | mp2an 417 | . . 3 ⊢ (1𝑜 +N 1𝑜) = (1𝑜 +𝑜 1𝑜) |
13 | 9, 12 | eleqtrri 2158 | . 2 ⊢ 1𝑜 ∈ (1𝑜 +N 1𝑜) |
14 | addclpi 6615 | . . . 4 ⊢ ((1𝑜 ∈ N ∧ 1𝑜 ∈ N) → (1𝑜 +N 1𝑜) ∈ N) | |
15 | 10, 10, 14 | mp2an 417 | . . 3 ⊢ (1𝑜 +N 1𝑜) ∈ N |
16 | ltpiord 6607 | . . 3 ⊢ ((1𝑜 ∈ N ∧ (1𝑜 +N 1𝑜) ∈ N) → (1𝑜 <N (1𝑜 +N 1𝑜) ↔ 1𝑜 ∈ (1𝑜 +N 1𝑜))) | |
17 | 10, 15, 16 | mp2an 417 | . 2 ⊢ (1𝑜 <N (1𝑜 +N 1𝑜) ↔ 1𝑜 ∈ (1𝑜 +N 1𝑜)) |
18 | 13, 17 | mpbir 144 | 1 ⊢ 1𝑜 <N (1𝑜 +N 1𝑜) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 = wceq 1285 ∈ wcel 1434 ∅c0 3268 class class class wbr 3806 ωcom 4360 (class class class)co 5564 1𝑜c1o 6079 +𝑜 coa 6083 Ncnpi 6560 +N cpli 6561 <N clti 6563 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3914 ax-sep 3917 ax-nul 3925 ax-pow 3969 ax-pr 3993 ax-un 4217 ax-setind 4309 ax-iinf 4358 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2612 df-sbc 2826 df-csb 2919 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-nul 3269 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-int 3658 df-iun 3701 df-br 3807 df-opab 3861 df-mpt 3862 df-tr 3897 df-eprel 4073 df-id 4077 df-iord 4150 df-on 4152 df-suc 4155 df-iom 4361 df-xp 4398 df-rel 4399 df-cnv 4400 df-co 4401 df-dm 4402 df-rn 4403 df-res 4404 df-ima 4405 df-iota 4918 df-fun 4955 df-fn 4956 df-f 4957 df-f1 4958 df-fo 4959 df-f1o 4960 df-fv 4961 df-ov 5567 df-oprab 5568 df-mpt2 5569 df-1st 5819 df-2nd 5820 df-recs 5975 df-irdg 6040 df-1o 6086 df-oadd 6090 df-ni 6592 df-pli 6593 df-lti 6595 |
This theorem is referenced by: 1lt2nq 6694 |
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