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Theorem ltpiord 6571
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 6559 . . 3 <N = ( E ∩ (N × N))
21breqi 3799 . 2 (𝐴 <N 𝐵𝐴( E ∩ (N × N))𝐵)
3 brinxp 4434 . . 3 ((𝐴N𝐵N) → (𝐴 E 𝐵𝐴( E ∩ (N × N))𝐵))
4 epelg 4053 . . . 4 (𝐵N → (𝐴 E 𝐵𝐴𝐵))
54adantl 271 . . 3 ((𝐴N𝐵N) → (𝐴 E 𝐵𝐴𝐵))
63, 5bitr3d 188 . 2 ((𝐴N𝐵N) → (𝐴( E ∩ (N × N))𝐵𝐴𝐵))
72, 6syl5bb 190 1 ((𝐴N𝐵N) → (𝐴 <N 𝐵𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wcel 1434  cin 2973   class class class wbr 3793   E cep 4050   × cxp 4369  Ncnpi 6524   <N clti 6527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-eprel 4052  df-xp 4377  df-lti 6559
This theorem is referenced by:  ltsopi  6572  pitric  6573  pitri3or  6574  ltdcpi  6575  ltexpi  6589  ltapig  6590  ltmpig  6591  1lt2pi  6592  nlt1pig  6593  archnqq  6669  prarloclemarch2  6671  prarloclemlt  6745  prarloclemn  6751
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