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Mirrors > Home > ILE Home > Th. List > prarloclemn | GIF version |
Description: Subtracting two from a positive integer. Lemma for prarloc 7279. (Contributed by Jim Kingdon, 5-Nov-2019.) |
Ref | Expression |
---|---|
prarloclemn | ⊢ ((𝑁 ∈ N ∧ 1o <N 𝑁) → ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . 3 ⊢ ((𝑁 ∈ N ∧ 1o <N 𝑁) → 𝑁 ∈ N) | |
2 | 1pi 7091 | . . . . 5 ⊢ 1o ∈ N | |
3 | ltpiord 7095 | . . . . 5 ⊢ ((1o ∈ N ∧ 𝑁 ∈ N) → (1o <N 𝑁 ↔ 1o ∈ 𝑁)) | |
4 | 2, 3 | mpan 420 | . . . 4 ⊢ (𝑁 ∈ N → (1o <N 𝑁 ↔ 1o ∈ 𝑁)) |
5 | 4 | biimpa 294 | . . 3 ⊢ ((𝑁 ∈ N ∧ 1o <N 𝑁) → 1o ∈ 𝑁) |
6 | piord 7087 | . . . 4 ⊢ (𝑁 ∈ N → Ord 𝑁) | |
7 | ordsucss 4390 | . . . 4 ⊢ (Ord 𝑁 → (1o ∈ 𝑁 → suc 1o ⊆ 𝑁)) | |
8 | 6, 7 | syl 14 | . . 3 ⊢ (𝑁 ∈ N → (1o ∈ 𝑁 → suc 1o ⊆ 𝑁)) |
9 | 1, 5, 8 | sylc 62 | . 2 ⊢ ((𝑁 ∈ N ∧ 1o <N 𝑁) → suc 1o ⊆ 𝑁) |
10 | df-2o 6282 | . . . 4 ⊢ 2o = suc 1o | |
11 | 10 | sseq1i 3093 | . . 3 ⊢ (2o ⊆ 𝑁 ↔ suc 1o ⊆ 𝑁) |
12 | pinn 7085 | . . . . 5 ⊢ (𝑁 ∈ N → 𝑁 ∈ ω) | |
13 | 2onn 6385 | . . . . . 6 ⊢ 2o ∈ ω | |
14 | nnawordex 6392 | . . . . . 6 ⊢ ((2o ∈ ω ∧ 𝑁 ∈ ω) → (2o ⊆ 𝑁 ↔ ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁)) | |
15 | 13, 14 | mpan 420 | . . . . 5 ⊢ (𝑁 ∈ ω → (2o ⊆ 𝑁 ↔ ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁)) |
16 | 12, 15 | syl 14 | . . . 4 ⊢ (𝑁 ∈ N → (2o ⊆ 𝑁 ↔ ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁)) |
17 | 16 | adantr 274 | . . 3 ⊢ ((𝑁 ∈ N ∧ 1o <N 𝑁) → (2o ⊆ 𝑁 ↔ ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁)) |
18 | 11, 17 | syl5bbr 193 | . 2 ⊢ ((𝑁 ∈ N ∧ 1o <N 𝑁) → (suc 1o ⊆ 𝑁 ↔ ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁)) |
19 | 9, 18 | mpbid 146 | 1 ⊢ ((𝑁 ∈ N ∧ 1o <N 𝑁) → ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1316 ∈ wcel 1465 ∃wrex 2394 ⊆ wss 3041 class class class wbr 3899 Ord word 4254 suc csuc 4257 ωcom 4474 (class class class)co 5742 1oc1o 6274 2oc2o 6275 +o coa 6278 Ncnpi 7048 <N clti 7051 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-eprel 4181 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-1o 6281 df-2o 6282 df-oadd 6285 df-ni 7080 df-lti 7083 |
This theorem is referenced by: prarloclem5 7276 |
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