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Theorem ltsopi 7096
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 4433 . . . . . 6 ¬ 𝑥𝑥
2 ltpiord 7095 . . . . . . 7 ((𝑥N𝑥N) → (𝑥 <N 𝑥𝑥𝑥))
32anidms 394 . . . . . 6 (𝑥N → (𝑥 <N 𝑥𝑥𝑥))
41, 3mtbiri 649 . . . . 5 (𝑥N → ¬ 𝑥 <N 𝑥)
54adantl 275 . . . 4 ((⊤ ∧ 𝑥N) → ¬ 𝑥 <N 𝑥)
6 pion 7086 . . . . . . . 8 (𝑧N𝑧 ∈ On)
7 ontr1 4281 . . . . . . . 8 (𝑧 ∈ On → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
86, 7syl 14 . . . . . . 7 (𝑧N → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
983ad2ant3 989 . . . . . 6 ((𝑥N𝑦N𝑧N) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
10 ltpiord 7095 . . . . . . . 8 ((𝑥N𝑦N) → (𝑥 <N 𝑦𝑥𝑦))
11103adant3 986 . . . . . . 7 ((𝑥N𝑦N𝑧N) → (𝑥 <N 𝑦𝑥𝑦))
12 ltpiord 7095 . . . . . . . 8 ((𝑦N𝑧N) → (𝑦 <N 𝑧𝑦𝑧))
13123adant1 984 . . . . . . 7 ((𝑥N𝑦N𝑧N) → (𝑦 <N 𝑧𝑦𝑧))
1411, 13anbi12d 464 . . . . . 6 ((𝑥N𝑦N𝑧N) → ((𝑥 <N 𝑦𝑦 <N 𝑧) ↔ (𝑥𝑦𝑦𝑧)))
15 ltpiord 7095 . . . . . . 7 ((𝑥N𝑧N) → (𝑥 <N 𝑧𝑥𝑧))
16153adant2 985 . . . . . 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 4196 . . 3 (⊤ → <N Po N)
20 pinn 7085 . . . . . 6 (𝑥N𝑥 ∈ ω)
21 pinn 7085 . . . . . 6 (𝑦N𝑦 ∈ ω)
22 nntri3or 6357 . . . . . 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 7095 . . . . . . 7 ((𝑦N𝑥N) → (𝑦 <N 𝑥𝑦𝑥))
2625ancoms 266 . . . . . 6 ((𝑥N𝑦N) → (𝑦 <N 𝑥𝑦𝑥))
2710, 24, 263orbi123d 1274 . . . . 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 4211 . 2 (⊤ → <N Or N)
3130mptru 1325 1 <N Or N
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  w3o 946  w3a 947  wtru 1317  wcel 1465   class class class wbr 3899   Or wor 4187  Oncon0 4255  ωcom 4474  Ncnpi 7048   <N clti 7051
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-tr 3997  df-eprel 4181  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-ni 7080  df-lti 7083
This theorem is referenced by:  ltsonq  7174
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