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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 6560 | . . . . 5 ⊢ 1o ∈ ω | |
2 | nna0 6514 | . . . . 5 ⊢ (1o ∈ ω → (1o +o ∅) = 1o) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (1o +o ∅) = 1o |
4 | 0lt1o 6480 | . . . . 5 ⊢ ∅ ∈ 1o | |
5 | peano1 4618 | . . . . . 6 ⊢ ∅ ∈ ω | |
6 | nnaord 6549 | . . . . . 6 ⊢ ((∅ ∈ ω ∧ 1o ∈ ω ∧ 1o ∈ ω) → (∅ ∈ 1o ↔ (1o +o ∅) ∈ (1o +o 1o))) | |
7 | 5, 1, 1, 6 | mp3an 1348 | . . . . 5 ⊢ (∅ ∈ 1o ↔ (1o +o ∅) ∈ (1o +o 1o)) |
8 | 4, 7 | mpbi 145 | . . . 4 ⊢ (1o +o ∅) ∈ (1o +o 1o) |
9 | 3, 8 | eqeltrri 2263 | . . 3 ⊢ 1o ∈ (1o +o 1o) |
10 | 1pi 7361 | . . . 4 ⊢ 1o ∈ N | |
11 | addpiord 7362 | . . . 4 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o +N 1o) = (1o +o 1o)) | |
12 | 10, 10, 11 | mp2an 426 | . . 3 ⊢ (1o +N 1o) = (1o +o 1o) |
13 | 9, 12 | eleqtrri 2265 | . 2 ⊢ 1o ∈ (1o +N 1o) |
14 | addclpi 7373 | . . . 4 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o +N 1o) ∈ N) | |
15 | 10, 10, 14 | mp2an 426 | . . 3 ⊢ (1o +N 1o) ∈ N |
16 | ltpiord 7365 | . . 3 ⊢ ((1o ∈ N ∧ (1o +N 1o) ∈ N) → (1o <N (1o +N 1o) ↔ 1o ∈ (1o +N 1o))) | |
17 | 10, 15, 16 | mp2an 426 | . 2 ⊢ (1o <N (1o +N 1o) ↔ 1o ∈ (1o +N 1o)) |
18 | 13, 17 | mpbir 146 | 1 ⊢ 1o <N (1o +N 1o) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1364 ∈ wcel 2160 ∅c0 3442 class class class wbr 4025 ωcom 4614 (class class class)co 5906 1oc1o 6449 +o coa 6453 Ncnpi 7318 +N cpli 7319 <N clti 7321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4140 ax-sep 4143 ax-nul 4151 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-iinf 4612 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2758 df-sbc 2982 df-csb 3077 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-nul 3443 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-iun 3910 df-br 4026 df-opab 4087 df-mpt 4088 df-tr 4124 df-eprel 4314 df-id 4318 df-iord 4391 df-on 4393 df-suc 4396 df-iom 4615 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-f1 5247 df-fo 5248 df-f1o 5249 df-fv 5250 df-ov 5909 df-oprab 5910 df-mpo 5911 df-1st 6180 df-2nd 6181 df-recs 6345 df-irdg 6410 df-1o 6456 df-oadd 6460 df-ni 7350 df-pli 7351 df-lti 7353 |
This theorem is referenced by: 1lt2nq 7452 |
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