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Theorem ltpiord 6303
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 ((A N B N) → (A <N BA B))

Proof of Theorem ltpiord
StepHypRef Expression
1 df-lti 6291 . . 3 <N = ( E ∩ (N × N))
21breqi 3761 . 2 (A <N BA( E ∩ (N × N))B)
3 brinxp 4351 . . 3 ((A N B N) → (A E BA( E ∩ (N × N))B))
4 epelg 4018 . . . 4 (B N → (A E BA B))
54adantl 262 . . 3 ((A N B N) → (A E BA B))
63, 5bitr3d 179 . 2 ((A N B N) → (A( E ∩ (N × N))BA B))
72, 6syl5bb 181 1 ((A N B N) → (A <N BA B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wcel 1390  cin 2910   class class class wbr 3755   E cep 4015   × cxp 4286  Ncnpi 6256   <N clti 6259
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-eprel 4017  df-xp 4294  df-lti 6291
This theorem is referenced by:  ltsopi  6304  pitric  6305  pitri3or  6306  ltdcpi  6307  ltexpi  6321  ltapig  6322  ltmpig  6323  1lt2pi  6324  nlt1pig  6325  archnqq  6400  prarloclemarch2  6402  prarloclemlt  6476  prarloclemn  6482
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