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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 6479 | . . . . 5 ⊢ 1o ∈ ω | |
2 | nna0 6433 | . . . . 5 ⊢ (1o ∈ ω → (1o +o ∅) = 1o) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (1o +o ∅) = 1o |
4 | 0lt1o 6399 | . . . . 5 ⊢ ∅ ∈ 1o | |
5 | peano1 4565 | . . . . . 6 ⊢ ∅ ∈ ω | |
6 | nnaord 6468 | . . . . . 6 ⊢ ((∅ ∈ ω ∧ 1o ∈ ω ∧ 1o ∈ ω) → (∅ ∈ 1o ↔ (1o +o ∅) ∈ (1o +o 1o))) | |
7 | 5, 1, 1, 6 | mp3an 1326 | . . . . 5 ⊢ (∅ ∈ 1o ↔ (1o +o ∅) ∈ (1o +o 1o)) |
8 | 4, 7 | mpbi 144 | . . . 4 ⊢ (1o +o ∅) ∈ (1o +o 1o) |
9 | 3, 8 | eqeltrri 2238 | . . 3 ⊢ 1o ∈ (1o +o 1o) |
10 | 1pi 7247 | . . . 4 ⊢ 1o ∈ N | |
11 | addpiord 7248 | . . . 4 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o +N 1o) = (1o +o 1o)) | |
12 | 10, 10, 11 | mp2an 423 | . . 3 ⊢ (1o +N 1o) = (1o +o 1o) |
13 | 9, 12 | eleqtrri 2240 | . 2 ⊢ 1o ∈ (1o +N 1o) |
14 | addclpi 7259 | . . . 4 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o +N 1o) ∈ N) | |
15 | 10, 10, 14 | mp2an 423 | . . 3 ⊢ (1o +N 1o) ∈ N |
16 | ltpiord 7251 | . . 3 ⊢ ((1o ∈ N ∧ (1o +N 1o) ∈ N) → (1o <N (1o +N 1o) ↔ 1o ∈ (1o +N 1o))) | |
17 | 10, 15, 16 | mp2an 423 | . 2 ⊢ (1o <N (1o +N 1o) ↔ 1o ∈ (1o +N 1o)) |
18 | 13, 17 | mpbir 145 | 1 ⊢ 1o <N (1o +N 1o) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1342 ∈ wcel 2135 ∅c0 3404 class class class wbr 3976 ωcom 4561 (class class class)co 5836 1oc1o 6368 +o coa 6372 Ncnpi 7204 +N cpli 7205 <N clti 7207 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-oadd 6379 df-ni 7236 df-pli 7237 df-lti 7239 |
This theorem is referenced by: 1lt2nq 7338 |
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