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Theorem ltpiord 10782
Description: Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
Assertion
Ref Expression
ltpiord ((𝐴N𝐵N) → (𝐴 <N 𝐵𝐴𝐵))

Proof of Theorem ltpiord
StepHypRef Expression
1 df-lti 10770 . . 3 <N = ( E ∩ (N × N))
21breqi 5110 . 2 (𝐴 <N 𝐵𝐴( E ∩ (N × N))𝐵)
3 brinxp 5709 . . 3 ((𝐴N𝐵N) → (𝐴 E 𝐵𝐴( E ∩ (N × N))𝐵))
4 epelg 5537 . . . 4 (𝐵N → (𝐴 E 𝐵𝐴𝐵))
54adantl 483 . . 3 ((𝐴N𝐵N) → (𝐴 E 𝐵𝐴𝐵))
63, 5bitr3d 281 . 2 ((𝐴N𝐵N) → (𝐴( E ∩ (N × N))𝐵𝐴𝐵))
72, 6bitrid 283 1 ((𝐴N𝐵N) → (𝐴 <N 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  cin 3908   class class class wbr 5104   E cep 5535   × cxp 5630  Ncnpi 10739   <N clti 10742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-eprel 5536  df-xp 5638  df-lti 10770
This theorem is referenced by:  ltexpi  10797  ltapi  10798  ltmpi  10799  1lt2pi  10800  nlt1pi  10801  indpi  10802  nqereu  10824
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