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Mirrors > Home > ILE Home > Th. List > prarloclemn | GIF version |
Description: Subtracting two from a positive integer. Lemma for prarloc 7499. (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 109 | . . 3 ⊢ ((𝑁 ∈ N ∧ 1o <N 𝑁) → 𝑁 ∈ N) | |
2 | 1pi 7311 | . . . . 5 ⊢ 1o ∈ N | |
3 | ltpiord 7315 | . . . . 5 ⊢ ((1o ∈ N ∧ 𝑁 ∈ N) → (1o <N 𝑁 ↔ 1o ∈ 𝑁)) | |
4 | 2, 3 | mpan 424 | . . . 4 ⊢ (𝑁 ∈ N → (1o <N 𝑁 ↔ 1o ∈ 𝑁)) |
5 | 4 | biimpa 296 | . . 3 ⊢ ((𝑁 ∈ N ∧ 1o <N 𝑁) → 1o ∈ 𝑁) |
6 | piord 7307 | . . . 4 ⊢ (𝑁 ∈ N → Ord 𝑁) | |
7 | ordsucss 4502 | . . . 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 6415 | . . . 4 ⊢ 2o = suc 1o | |
11 | 10 | sseq1i 3181 | . . 3 ⊢ (2o ⊆ 𝑁 ↔ suc 1o ⊆ 𝑁) |
12 | pinn 7305 | . . . . 5 ⊢ (𝑁 ∈ N → 𝑁 ∈ ω) | |
13 | 2onn 6519 | . . . . . 6 ⊢ 2o ∈ ω | |
14 | nnawordex 6527 | . . . . . 6 ⊢ ((2o ∈ ω ∧ 𝑁 ∈ ω) → (2o ⊆ 𝑁 ↔ ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁)) | |
15 | 13, 14 | mpan 424 | . . . . 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 276 | . . 3 ⊢ ((𝑁 ∈ N ∧ 1o <N 𝑁) → (2o ⊆ 𝑁 ↔ ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁)) |
18 | 11, 17 | bitr3id 194 | . 2 ⊢ ((𝑁 ∈ N ∧ 1o <N 𝑁) → (suc 1o ⊆ 𝑁 ↔ ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁)) |
19 | 9, 18 | mpbid 147 | 1 ⊢ ((𝑁 ∈ N ∧ 1o <N 𝑁) → ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 ⊆ wss 3129 class class class wbr 4002 Ord word 4361 suc csuc 4364 ωcom 4588 (class class class)co 5872 1oc1o 6407 2oc2o 6408 +o coa 6411 Ncnpi 7268 <N clti 7271 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-eprel 4288 df-id 4292 df-iord 4365 df-on 4367 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-ov 5875 df-oprab 5876 df-mpo 5877 df-1st 6138 df-2nd 6139 df-recs 6303 df-irdg 6368 df-1o 6414 df-2o 6415 df-oadd 6418 df-ni 7300 df-lti 7303 |
This theorem is referenced by: prarloclem5 7496 |
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