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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 8254 | . . . . 5 ⊢ 1o ∈ ω | |
2 | nna0 8219 | . . . . 5 ⊢ (1o ∈ ω → (1o +o ∅) = 1o) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (1o +o ∅) = 1o |
4 | 0lt1o 8118 | . . . . 5 ⊢ ∅ ∈ 1o | |
5 | peano1 7590 | . . . . . 6 ⊢ ∅ ∈ ω | |
6 | nnaord 8234 | . . . . . 6 ⊢ ((∅ ∈ ω ∧ 1o ∈ ω ∧ 1o ∈ ω) → (∅ ∈ 1o ↔ (1o +o ∅) ∈ (1o +o 1o))) | |
7 | 5, 1, 1, 6 | mp3an 1452 | . . . . 5 ⊢ (∅ ∈ 1o ↔ (1o +o ∅) ∈ (1o +o 1o)) |
8 | 4, 7 | mpbi 231 | . . . 4 ⊢ (1o +o ∅) ∈ (1o +o 1o) |
9 | 3, 8 | eqeltrri 2907 | . . 3 ⊢ 1o ∈ (1o +o 1o) |
10 | 1pi 10293 | . . . 4 ⊢ 1o ∈ N | |
11 | addpiord 10294 | . . . 4 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o +N 1o) = (1o +o 1o)) | |
12 | 10, 10, 11 | mp2an 688 | . . 3 ⊢ (1o +N 1o) = (1o +o 1o) |
13 | 9, 12 | eleqtrri 2909 | . 2 ⊢ 1o ∈ (1o +N 1o) |
14 | addclpi 10302 | . . . 4 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o +N 1o) ∈ N) | |
15 | 10, 10, 14 | mp2an 688 | . . 3 ⊢ (1o +N 1o) ∈ N |
16 | ltpiord 10297 | . . 3 ⊢ ((1o ∈ N ∧ (1o +N 1o) ∈ N) → (1o <N (1o +N 1o) ↔ 1o ∈ (1o +N 1o))) | |
17 | 10, 15, 16 | mp2an 688 | . 2 ⊢ (1o <N (1o +N 1o) ↔ 1o ∈ (1o +N 1o)) |
18 | 13, 17 | mpbir 232 | 1 ⊢ 1o <N (1o +N 1o) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 ∅c0 4288 class class class wbr 5057 (class class class)co 7145 ωcom 7569 1oc1o 8084 +o coa 8088 Ncnpi 10254 +N cpli 10255 <N clti 10257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-ni 10282 df-pli 10283 df-lti 10285 |
This theorem is referenced by: 1lt2nq 10383 |
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