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Theorem ltpiord 6417
Description: Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
ltpiord ((𝐴N𝐵N) → (𝐴 <N 𝐵𝐴𝐵))

Proof of Theorem ltpiord
StepHypRef Expression
1 df-lti 6405 . . 3 <N = ( E ∩ (N × N))
21breqi 3770 . 2 (𝐴 <N 𝐵𝐴( E ∩ (N × N))𝐵)
3 brinxp 4408 . . 3 ((𝐴N𝐵N) → (𝐴 E 𝐵𝐴( E ∩ (N × N))𝐵))
4 epelg 4027 . . . 4 (𝐵N → (𝐴 E 𝐵𝐴𝐵))
54adantl 262 . . 3 ((𝐴N𝐵N) → (𝐴 E 𝐵𝐴𝐵))
63, 5bitr3d 179 . 2 ((𝐴N𝐵N) → (𝐴( E ∩ (N × N))𝐵𝐴𝐵))
72, 6syl5bb 181 1 ((𝐴N𝐵N) → (𝐴 <N 𝐵𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wcel 1393  cin 2916   class class class wbr 3764   E cep 4024   × cxp 4343  Ncnpi 6370   <N clti 6373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-eprel 4026  df-xp 4351  df-lti 6405
This theorem is referenced by:  ltsopi  6418  pitric  6419  pitri3or  6420  ltdcpi  6421  ltexpi  6435  ltapig  6436  ltmpig  6437  1lt2pi  6438  nlt1pig  6439  archnqq  6515  prarloclemarch2  6517  prarloclemlt  6591  prarloclemn  6597
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