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Mirrors > Home > ILE Home > Th. List > prarloclemn | GIF version |
Description: Subtracting two from a positive integer. Lemma for prarloc 7311. (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 7123 | . . . . 5 ⊢ 1o ∈ N | |
3 | ltpiord 7127 | . . . . 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 7119 | . . . 4 ⊢ (𝑁 ∈ N → Ord 𝑁) | |
7 | ordsucss 4420 | . . . 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 6314 | . . . 4 ⊢ 2o = suc 1o | |
11 | 10 | sseq1i 3123 | . . 3 ⊢ (2o ⊆ 𝑁 ↔ suc 1o ⊆ 𝑁) |
12 | pinn 7117 | . . . . 5 ⊢ (𝑁 ∈ N → 𝑁 ∈ ω) | |
13 | 2onn 6417 | . . . . . 6 ⊢ 2o ∈ ω | |
14 | nnawordex 6424 | . . . . . 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 1331 ∈ wcel 1480 ∃wrex 2417 ⊆ wss 3071 class class class wbr 3929 Ord word 4284 suc csuc 4287 ωcom 4504 (class class class)co 5774 1oc1o 6306 2oc2o 6307 +o coa 6310 Ncnpi 7080 <N clti 7083 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-2o 6314 df-oadd 6317 df-ni 7112 df-lti 7115 |
This theorem is referenced by: prarloclem5 7308 |
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