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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 6688 | . . . . 5 ⊢ 1o ∈ ω | |
| 2 | nna0 6642 | . . . . 5 ⊢ (1o ∈ ω → (1o +o ∅) = 1o) | |
| 3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (1o +o ∅) = 1o |
| 4 | 0lt1o 6608 | . . . . 5 ⊢ ∅ ∈ 1o | |
| 5 | peano1 4692 | . . . . . 6 ⊢ ∅ ∈ ω | |
| 6 | nnaord 6677 | . . . . . 6 ⊢ ((∅ ∈ ω ∧ 1o ∈ ω ∧ 1o ∈ ω) → (∅ ∈ 1o ↔ (1o +o ∅) ∈ (1o +o 1o))) | |
| 7 | 5, 1, 1, 6 | mp3an 1373 | . . . . 5 ⊢ (∅ ∈ 1o ↔ (1o +o ∅) ∈ (1o +o 1o)) |
| 8 | 4, 7 | mpbi 145 | . . . 4 ⊢ (1o +o ∅) ∈ (1o +o 1o) |
| 9 | 3, 8 | eqeltrri 2305 | . . 3 ⊢ 1o ∈ (1o +o 1o) |
| 10 | 1pi 7535 | . . . 4 ⊢ 1o ∈ N | |
| 11 | addpiord 7536 | . . . 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 2307 | . 2 ⊢ 1o ∈ (1o +N 1o) |
| 14 | addclpi 7547 | . . . 4 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o +N 1o) ∈ N) | |
| 15 | 10, 10, 14 | mp2an 426 | . . 3 ⊢ (1o +N 1o) ∈ N |
| 16 | ltpiord 7539 | . . 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 1397 ∈ wcel 2202 ∅c0 3494 class class class wbr 4088 ωcom 4688 (class class class)co 6018 1oc1o 6575 +o coa 6579 Ncnpi 7492 +N cpli 7493 <N clti 7495 |
| 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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 |
| This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-eprel 4386 df-id 4390 df-iord 4463 df-on 4465 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-recs 6471 df-irdg 6536 df-1o 6582 df-oadd 6586 df-ni 7524 df-pli 7525 df-lti 7527 |
| This theorem is referenced by: 1lt2nq 7626 |
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