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Theorem ltsopi 6933
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 4377 . . . . . 6 ¬ 𝑥𝑥
2 ltpiord 6932 . . . . . . 7 ((𝑥N𝑥N) → (𝑥 <N 𝑥𝑥𝑥))
32anidms 390 . . . . . 6 (𝑥N → (𝑥 <N 𝑥𝑥𝑥))
41, 3mtbiri 636 . . . . 5 (𝑥N → ¬ 𝑥 <N 𝑥)
54adantl 272 . . . 4 ((⊤ ∧ 𝑥N) → ¬ 𝑥 <N 𝑥)
6 pion 6923 . . . . . . . 8 (𝑧N𝑧 ∈ On)
7 ontr1 4225 . . . . . . . 8 (𝑧 ∈ On → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
86, 7syl 14 . . . . . . 7 (𝑧N → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
983ad2ant3 967 . . . . . 6 ((𝑥N𝑦N𝑧N) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
10 ltpiord 6932 . . . . . . . 8 ((𝑥N𝑦N) → (𝑥 <N 𝑦𝑥𝑦))
11103adant3 964 . . . . . . 7 ((𝑥N𝑦N𝑧N) → (𝑥 <N 𝑦𝑥𝑦))
12 ltpiord 6932 . . . . . . . 8 ((𝑦N𝑧N) → (𝑦 <N 𝑧𝑦𝑧))
13123adant1 962 . . . . . . 7 ((𝑥N𝑦N𝑧N) → (𝑦 <N 𝑧𝑦𝑧))
1411, 13anbi12d 458 . . . . . 6 ((𝑥N𝑦N𝑧N) → ((𝑥 <N 𝑦𝑦 <N 𝑧) ↔ (𝑥𝑦𝑦𝑧)))
15 ltpiord 6932 . . . . . . 7 ((𝑥N𝑧N) → (𝑥 <N 𝑧𝑥𝑧))
16153adant2 963 . . . . . 6 ((𝑥N𝑦N𝑧N) → (𝑥 <N 𝑧𝑥𝑧))
179, 14, 163imtr4d 202 . . . . 5 ((𝑥N𝑦N𝑧N) → ((𝑥 <N 𝑦𝑦 <N 𝑧) → 𝑥 <N 𝑧))
1817adantl 272 . . . 4 ((⊤ ∧ (𝑥N𝑦N𝑧N)) → ((𝑥 <N 𝑦𝑦 <N 𝑧) → 𝑥 <N 𝑧))
195, 18ispod 4140 . . 3 (⊤ → <N Po N)
20 pinn 6922 . . . . . 6 (𝑥N𝑥 ∈ ω)
21 pinn 6922 . . . . . 6 (𝑦N𝑦 ∈ ω)
22 nntri3or 6268 . . . . . 6 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
2320, 21, 22syl2an 284 . . . . 5 ((𝑥N𝑦N) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
24 biidd 171 . . . . . 6 ((𝑥N𝑦N) → (𝑥 = 𝑦𝑥 = 𝑦))
25 ltpiord 6932 . . . . . . 7 ((𝑦N𝑥N) → (𝑦 <N 𝑥𝑦𝑥))
2625ancoms 265 . . . . . 6 ((𝑥N𝑦N) → (𝑦 <N 𝑥𝑦𝑥))
2710, 24, 263orbi123d 1248 . . . . 5 ((𝑥N𝑦N) → ((𝑥 <N 𝑦𝑥 = 𝑦𝑦 <N 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
2823, 27mpbird 166 . . . 4 ((𝑥N𝑦N) → (𝑥 <N 𝑦𝑥 = 𝑦𝑦 <N 𝑥))
2928adantl 272 . . 3 ((⊤ ∧ (𝑥N𝑦N)) → (𝑥 <N 𝑦𝑥 = 𝑦𝑦 <N 𝑥))
3019, 29issod 4155 . 2 (⊤ → <N Or N)
3130mptru 1299 1 <N Or N
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  w3o 924  w3a 925  wtru 1291  wcel 1439   class class class wbr 3851   Or wor 4131  Oncon0 4199  ωcom 4418  Ncnpi 6885   <N clti 6888
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-br 3852  df-opab 3906  df-tr 3943  df-eprel 4125  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4457  df-ni 6917  df-lti 6920
This theorem is referenced by:  ltsonq  7011
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