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Theorem ltsopi 6418
 Description: Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.)
Assertion
Ref Expression
ltsopi <N Or N

Proof of Theorem ltsopi
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elirrv 4272 . . . . . 6 ¬ 𝑥𝑥
2 ltpiord 6417 . . . . . . 7 ((𝑥N𝑥N) → (𝑥 <N 𝑥𝑥𝑥))
32anidms 377 . . . . . 6 (𝑥N → (𝑥 <N 𝑥𝑥𝑥))
41, 3mtbiri 600 . . . . 5 (𝑥N → ¬ 𝑥 <N 𝑥)
54adantl 262 . . . 4 ((⊤ ∧ 𝑥N) → ¬ 𝑥 <N 𝑥)
6 pion 6408 . . . . . . . 8 (𝑧N𝑧 ∈ On)
7 ontr1 4126 . . . . . . . 8 (𝑧 ∈ On → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
86, 7syl 14 . . . . . . 7 (𝑧N → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
983ad2ant3 927 . . . . . 6 ((𝑥N𝑦N𝑧N) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
10 ltpiord 6417 . . . . . . . 8 ((𝑥N𝑦N) → (𝑥 <N 𝑦𝑥𝑦))
11103adant3 924 . . . . . . 7 ((𝑥N𝑦N𝑧N) → (𝑥 <N 𝑦𝑥𝑦))
12 ltpiord 6417 . . . . . . . 8 ((𝑦N𝑧N) → (𝑦 <N 𝑧𝑦𝑧))
13123adant1 922 . . . . . . 7 ((𝑥N𝑦N𝑧N) → (𝑦 <N 𝑧𝑦𝑧))
1411, 13anbi12d 442 . . . . . 6 ((𝑥N𝑦N𝑧N) → ((𝑥 <N 𝑦𝑦 <N 𝑧) ↔ (𝑥𝑦𝑦𝑧)))
15 ltpiord 6417 . . . . . . 7 ((𝑥N𝑧N) → (𝑥 <N 𝑧𝑥𝑧))
16153adant2 923 . . . . . 6 ((𝑥N𝑦N𝑧N) → (𝑥 <N 𝑧𝑥𝑧))
179, 14, 163imtr4d 192 . . . . 5 ((𝑥N𝑦N𝑧N) → ((𝑥 <N 𝑦𝑦 <N 𝑧) → 𝑥 <N 𝑧))
1817adantl 262 . . . 4 ((⊤ ∧ (𝑥N𝑦N𝑧N)) → ((𝑥 <N 𝑦𝑦 <N 𝑧) → 𝑥 <N 𝑧))
195, 18ispod 4041 . . 3 (⊤ → <N Po N)
20 pinn 6407 . . . . . 6 (𝑥N𝑥 ∈ ω)
21 pinn 6407 . . . . . 6 (𝑦N𝑦 ∈ ω)
22 nntri3or 6072 . . . . . 6 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
2320, 21, 22syl2an 273 . . . . 5 ((𝑥N𝑦N) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
24 biidd 161 . . . . . 6 ((𝑥N𝑦N) → (𝑥 = 𝑦𝑥 = 𝑦))
25 ltpiord 6417 . . . . . . 7 ((𝑦N𝑥N) → (𝑦 <N 𝑥𝑦𝑥))
2625ancoms 255 . . . . . 6 ((𝑥N𝑦N) → (𝑦 <N 𝑥𝑦𝑥))
2710, 24, 263orbi123d 1206 . . . . 5 ((𝑥N𝑦N) → ((𝑥 <N 𝑦𝑥 = 𝑦𝑦 <N 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
2823, 27mpbird 156 . . . 4 ((𝑥N𝑦N) → (𝑥 <N 𝑦𝑥 = 𝑦𝑦 <N 𝑥))
2928adantl 262 . . 3 ((⊤ ∧ (𝑥N𝑦N)) → (𝑥 <N 𝑦𝑥 = 𝑦𝑦 <N 𝑥))
3019, 29issod 4056 . 2 (⊤ → <N Or N)
3130trud 1252 1 <N Or N
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ w3o 884   ∧ w3a 885  ⊤wtru 1244   ∈ wcel 1393   class class class wbr 3764   Or wor 4032  Oncon0 4100  ωcom 4313  Ncnpi 6370
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