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Theorem ltsopi 7092
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 4431 . . . . . 6 ¬ 𝑥𝑥
2 ltpiord 7091 . . . . . . 7 ((𝑥N𝑥N) → (𝑥 <N 𝑥𝑥𝑥))
32anidms 392 . . . . . 6 (𝑥N → (𝑥 <N 𝑥𝑥𝑥))
41, 3mtbiri 647 . . . . 5 (𝑥N → ¬ 𝑥 <N 𝑥)
54adantl 273 . . . 4 ((⊤ ∧ 𝑥N) → ¬ 𝑥 <N 𝑥)
6 pion 7082 . . . . . . . 8 (𝑧N𝑧 ∈ On)
7 ontr1 4279 . . . . . . . 8 (𝑧 ∈ On → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
86, 7syl 14 . . . . . . 7 (𝑧N → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
983ad2ant3 987 . . . . . 6 ((𝑥N𝑦N𝑧N) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
10 ltpiord 7091 . . . . . . . 8 ((𝑥N𝑦N) → (𝑥 <N 𝑦𝑥𝑦))
11103adant3 984 . . . . . . 7 ((𝑥N𝑦N𝑧N) → (𝑥 <N 𝑦𝑥𝑦))
12 ltpiord 7091 . . . . . . . 8 ((𝑦N𝑧N) → (𝑦 <N 𝑧𝑦𝑧))
13123adant1 982 . . . . . . 7 ((𝑥N𝑦N𝑧N) → (𝑦 <N 𝑧𝑦𝑧))
1411, 13anbi12d 462 . . . . . 6 ((𝑥N𝑦N𝑧N) → ((𝑥 <N 𝑦𝑦 <N 𝑧) ↔ (𝑥𝑦𝑦𝑧)))
15 ltpiord 7091 . . . . . . 7 ((𝑥N𝑧N) → (𝑥 <N 𝑧𝑥𝑧))
16153adant2 983 . . . . . 6 ((𝑥N𝑦N𝑧N) → (𝑥 <N 𝑧𝑥𝑧))
179, 14, 163imtr4d 202 . . . . 5 ((𝑥N𝑦N𝑧N) → ((𝑥 <N 𝑦𝑦 <N 𝑧) → 𝑥 <N 𝑧))
1817adantl 273 . . . 4 ((⊤ ∧ (𝑥N𝑦N𝑧N)) → ((𝑥 <N 𝑦𝑦 <N 𝑧) → 𝑥 <N 𝑧))
195, 18ispod 4194 . . 3 (⊤ → <N Po N)
20 pinn 7081 . . . . . 6 (𝑥N𝑥 ∈ ω)
21 pinn 7081 . . . . . 6 (𝑦N𝑦 ∈ ω)
22 nntri3or 6355 . . . . . 6 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
2320, 21, 22syl2an 285 . . . . 5 ((𝑥N𝑦N) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
24 biidd 171 . . . . . 6 ((𝑥N𝑦N) → (𝑥 = 𝑦𝑥 = 𝑦))
25 ltpiord 7091 . . . . . . 7 ((𝑦N𝑥N) → (𝑦 <N 𝑥𝑦𝑥))
2625ancoms 266 . . . . . 6 ((𝑥N𝑦N) → (𝑦 <N 𝑥𝑦𝑥))
2710, 24, 263orbi123d 1272 . . . . 5 ((𝑥N𝑦N) → ((𝑥 <N 𝑦𝑥 = 𝑦𝑦 <N 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
2823, 27mpbird 166 . . . 4 ((𝑥N𝑦N) → (𝑥 <N 𝑦𝑥 = 𝑦𝑦 <N 𝑥))
2928adantl 273 . . 3 ((⊤ ∧ (𝑥N𝑦N)) → (𝑥 <N 𝑦𝑥 = 𝑦𝑦 <N 𝑥))
3019, 29issod 4209 . 2 (⊤ → <N Or N)
3130mptru 1323 1 <N Or N
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  w3o 944  w3a 945  wtru 1315  wcel 1463   class class class wbr 3897   Or wor 4185  Oncon0 4253  ωcom 4472  Ncnpi 7044   <N clti 7047
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-tr 3995  df-eprel 4179  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-ni 7076  df-lti 7079
This theorem is referenced by:  ltsonq  7170
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