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Theorem ltsopi 7282
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 4532 . . . . . 6 ¬ 𝑥𝑥
2 ltpiord 7281 . . . . . . 7 ((𝑥N𝑥N) → (𝑥 <N 𝑥𝑥𝑥))
32anidms 395 . . . . . 6 (𝑥N → (𝑥 <N 𝑥𝑥𝑥))
41, 3mtbiri 670 . . . . 5 (𝑥N → ¬ 𝑥 <N 𝑥)
54adantl 275 . . . 4 ((⊤ ∧ 𝑥N) → ¬ 𝑥 <N 𝑥)
6 pion 7272 . . . . . . . 8 (𝑧N𝑧 ∈ On)
7 ontr1 4374 . . . . . . . 8 (𝑧 ∈ On → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
86, 7syl 14 . . . . . . 7 (𝑧N → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
983ad2ant3 1015 . . . . . 6 ((𝑥N𝑦N𝑧N) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
10 ltpiord 7281 . . . . . . . 8 ((𝑥N𝑦N) → (𝑥 <N 𝑦𝑥𝑦))
11103adant3 1012 . . . . . . 7 ((𝑥N𝑦N𝑧N) → (𝑥 <N 𝑦𝑥𝑦))
12 ltpiord 7281 . . . . . . . 8 ((𝑦N𝑧N) → (𝑦 <N 𝑧𝑦𝑧))
13123adant1 1010 . . . . . . 7 ((𝑥N𝑦N𝑧N) → (𝑦 <N 𝑧𝑦𝑧))
1411, 13anbi12d 470 . . . . . 6 ((𝑥N𝑦N𝑧N) → ((𝑥 <N 𝑦𝑦 <N 𝑧) ↔ (𝑥𝑦𝑦𝑧)))
15 ltpiord 7281 . . . . . . 7 ((𝑥N𝑧N) → (𝑥 <N 𝑧𝑥𝑧))
16153adant2 1011 . . . . . 6 ((𝑥N𝑦N𝑧N) → (𝑥 <N 𝑧𝑥𝑧))
179, 14, 163imtr4d 202 . . . . 5 ((𝑥N𝑦N𝑧N) → ((𝑥 <N 𝑦𝑦 <N 𝑧) → 𝑥 <N 𝑧))
1817adantl 275 . . . 4 ((⊤ ∧ (𝑥N𝑦N𝑧N)) → ((𝑥 <N 𝑦𝑦 <N 𝑧) → 𝑥 <N 𝑧))
195, 18ispod 4289 . . 3 (⊤ → <N Po N)
20 pinn 7271 . . . . . 6 (𝑥N𝑥 ∈ ω)
21 pinn 7271 . . . . . 6 (𝑦N𝑦 ∈ ω)
22 nntri3or 6472 . . . . . 6 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
2320, 21, 22syl2an 287 . . . . 5 ((𝑥N𝑦N) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
24 biidd 171 . . . . . 6 ((𝑥N𝑦N) → (𝑥 = 𝑦𝑥 = 𝑦))
25 ltpiord 7281 . . . . . . 7 ((𝑦N𝑥N) → (𝑦 <N 𝑥𝑦𝑥))
2625ancoms 266 . . . . . 6 ((𝑥N𝑦N) → (𝑦 <N 𝑥𝑦𝑥))
2710, 24, 263orbi123d 1306 . . . . 5 ((𝑥N𝑦N) → ((𝑥 <N 𝑦𝑥 = 𝑦𝑦 <N 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
2823, 27mpbird 166 . . . 4 ((𝑥N𝑦N) → (𝑥 <N 𝑦𝑥 = 𝑦𝑦 <N 𝑥))
2928adantl 275 . . 3 ((⊤ ∧ (𝑥N𝑦N)) → (𝑥 <N 𝑦𝑥 = 𝑦𝑦 <N 𝑥))
3019, 29issod 4304 . 2 (⊤ → <N Or N)
3130mptru 1357 1 <N Or N
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  w3o 972  w3a 973  wtru 1349  wcel 2141   class class class wbr 3989   Or wor 4280  Oncon0 4348  ωcom 4574  Ncnpi 7234   <N clti 7237
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-tr 4088  df-eprel 4274  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-ni 7266  df-lti 7269
This theorem is referenced by:  ltsonq  7360
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