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Mirrors > Home > ILE Home > Th. List > prarloclemn | GIF version |
Description: Subtracting two from a positive integer. Lemma for prarloc 7435. (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 7247 | . . . . 5 ⊢ 1o ∈ N | |
3 | ltpiord 7251 | . . . . 5 ⊢ ((1o ∈ N ∧ 𝑁 ∈ N) → (1o <N 𝑁 ↔ 1o ∈ 𝑁)) | |
4 | 2, 3 | mpan 421 | . . . 4 ⊢ (𝑁 ∈ N → (1o <N 𝑁 ↔ 1o ∈ 𝑁)) |
5 | 4 | biimpa 294 | . . 3 ⊢ ((𝑁 ∈ N ∧ 1o <N 𝑁) → 1o ∈ 𝑁) |
6 | piord 7243 | . . . 4 ⊢ (𝑁 ∈ N → Ord 𝑁) | |
7 | ordsucss 4475 | . . . 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 6376 | . . . 4 ⊢ 2o = suc 1o | |
11 | 10 | sseq1i 3163 | . . 3 ⊢ (2o ⊆ 𝑁 ↔ suc 1o ⊆ 𝑁) |
12 | pinn 7241 | . . . . 5 ⊢ (𝑁 ∈ N → 𝑁 ∈ ω) | |
13 | 2onn 6480 | . . . . . 6 ⊢ 2o ∈ ω | |
14 | nnawordex 6487 | . . . . . 6 ⊢ ((2o ∈ ω ∧ 𝑁 ∈ ω) → (2o ⊆ 𝑁 ↔ ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁)) | |
15 | 13, 14 | mpan 421 | . . . . 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 | bitr3id 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 1342 ∈ wcel 2135 ∃wrex 2443 ⊆ wss 3111 class class class wbr 3976 Ord word 4334 suc csuc 4337 ωcom 4561 (class class class)co 5836 1oc1o 6368 2oc2o 6369 +o coa 6372 Ncnpi 7204 <N clti 7207 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-2o 6376 df-oadd 6379 df-ni 7236 df-lti 7239 |
This theorem is referenced by: prarloclem5 7432 |
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