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Mirrors > Home > MPE Home > Th. List > 1lt2pi | Structured version Visualization version GIF version |
Description: One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
1lt2pi | ⊢ 1𝑜 <N (1𝑜 +N 1𝑜) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 7888 | . . . . 5 ⊢ 1𝑜 ∈ ω | |
2 | nna0 7853 | . . . . 5 ⊢ (1𝑜 ∈ ω → (1𝑜 +𝑜 ∅) = 1𝑜) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (1𝑜 +𝑜 ∅) = 1𝑜 |
4 | 0lt1o 7753 | . . . . 5 ⊢ ∅ ∈ 1𝑜 | |
5 | peano1 7250 | . . . . . 6 ⊢ ∅ ∈ ω | |
6 | nnaord 7868 | . . . . . 6 ⊢ ((∅ ∈ ω ∧ 1𝑜 ∈ ω ∧ 1𝑜 ∈ ω) → (∅ ∈ 1𝑜 ↔ (1𝑜 +𝑜 ∅) ∈ (1𝑜 +𝑜 1𝑜))) | |
7 | 5, 1, 1, 6 | mp3an 1573 | . . . . 5 ⊢ (∅ ∈ 1𝑜 ↔ (1𝑜 +𝑜 ∅) ∈ (1𝑜 +𝑜 1𝑜)) |
8 | 4, 7 | mpbi 220 | . . . 4 ⊢ (1𝑜 +𝑜 ∅) ∈ (1𝑜 +𝑜 1𝑜) |
9 | 3, 8 | eqeltrri 2836 | . . 3 ⊢ 1𝑜 ∈ (1𝑜 +𝑜 1𝑜) |
10 | 1pi 9897 | . . . 4 ⊢ 1𝑜 ∈ N | |
11 | addpiord 9898 | . . . 4 ⊢ ((1𝑜 ∈ N ∧ 1𝑜 ∈ N) → (1𝑜 +N 1𝑜) = (1𝑜 +𝑜 1𝑜)) | |
12 | 10, 10, 11 | mp2an 710 | . . 3 ⊢ (1𝑜 +N 1𝑜) = (1𝑜 +𝑜 1𝑜) |
13 | 9, 12 | eleqtrri 2838 | . 2 ⊢ 1𝑜 ∈ (1𝑜 +N 1𝑜) |
14 | addclpi 9906 | . . . 4 ⊢ ((1𝑜 ∈ N ∧ 1𝑜 ∈ N) → (1𝑜 +N 1𝑜) ∈ N) | |
15 | 10, 10, 14 | mp2an 710 | . . 3 ⊢ (1𝑜 +N 1𝑜) ∈ N |
16 | ltpiord 9901 | . . 3 ⊢ ((1𝑜 ∈ N ∧ (1𝑜 +N 1𝑜) ∈ N) → (1𝑜 <N (1𝑜 +N 1𝑜) ↔ 1𝑜 ∈ (1𝑜 +N 1𝑜))) | |
17 | 10, 15, 16 | mp2an 710 | . 2 ⊢ (1𝑜 <N (1𝑜 +N 1𝑜) ↔ 1𝑜 ∈ (1𝑜 +N 1𝑜)) |
18 | 13, 17 | mpbir 221 | 1 ⊢ 1𝑜 <N (1𝑜 +N 1𝑜) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1632 ∈ wcel 2139 ∅c0 4058 class class class wbr 4804 (class class class)co 6813 ωcom 7230 1𝑜c1o 7722 +𝑜 coa 7726 Ncnpi 9858 +N cpli 9859 <N clti 9861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-ni 9886 df-pli 9887 df-lti 9889 |
This theorem is referenced by: 1lt2nq 9987 |
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