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Theorem prarloclemn 7332
 Description: Subtracting two from a positive integer. Lemma for prarloc 7336. (Contributed by Jim Kingdon, 5-Nov-2019.)
Assertion
Ref Expression
prarloclemn ((𝑁N ∧ 1o <N 𝑁) → ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁)
Distinct variable group:   𝑥,𝑁

Proof of Theorem prarloclemn
StepHypRef Expression
1 simpl 108 . . 3 ((𝑁N ∧ 1o <N 𝑁) → 𝑁N)
2 1pi 7148 . . . . 5 1oN
3 ltpiord 7152 . . . . 5 ((1oN𝑁N) → (1o <N 𝑁 ↔ 1o𝑁))
42, 3mpan 421 . . . 4 (𝑁N → (1o <N 𝑁 ↔ 1o𝑁))
54biimpa 294 . . 3 ((𝑁N ∧ 1o <N 𝑁) → 1o𝑁)
6 piord 7144 . . . 4 (𝑁N → Ord 𝑁)
7 ordsucss 4428 . . . 4 (Ord 𝑁 → (1o𝑁 → suc 1o𝑁))
86, 7syl 14 . . 3 (𝑁N → (1o𝑁 → suc 1o𝑁))
91, 5, 8sylc 62 . 2 ((𝑁N ∧ 1o <N 𝑁) → suc 1o𝑁)
10 df-2o 6322 . . . 4 2o = suc 1o
1110sseq1i 3128 . . 3 (2o𝑁 ↔ suc 1o𝑁)
12 pinn 7142 . . . . 5 (𝑁N𝑁 ∈ ω)
13 2onn 6425 . . . . . 6 2o ∈ ω
14 nnawordex 6432 . . . . . 6 ((2o ∈ ω ∧ 𝑁 ∈ ω) → (2o𝑁 ↔ ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁))
1513, 14mpan 421 . . . . 5 (𝑁 ∈ ω → (2o𝑁 ↔ ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁))
1612, 15syl 14 . . . 4 (𝑁N → (2o𝑁 ↔ ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁))
1716adantr 274 . . 3 ((𝑁N ∧ 1o <N 𝑁) → (2o𝑁 ↔ ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁))
1811, 17bitr3id 193 . 2 ((𝑁N ∧ 1o <N 𝑁) → (suc 1o𝑁 ↔ ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁))
199, 18mpbid 146 1 ((𝑁N ∧ 1o <N 𝑁) → ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  ∃wrex 2418   ⊆ wss 3076   class class class wbr 3937  Ord word 4292  suc csuc 4295  ωcom 4512  (class class class)co 5782  1oc1o 6314  2oc2o 6315   +o coa 6318  Ncnpi 7105
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