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Theorem ltpiord 10927
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 10915 . . 3 <N = ( E ∩ (N × N))
21breqi 5149 . 2 (𝐴 <N 𝐵𝐴( E ∩ (N × N))𝐵)
3 brinxp 5764 . . 3 ((𝐴N𝐵N) → (𝐴 E 𝐵𝐴( E ∩ (N × N))𝐵))
4 epelg 5585 . . . 4 (𝐵N → (𝐴 E 𝐵𝐴𝐵))
54adantl 481 . . 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 206  wa 395  wcel 2108  cin 3950   class class class wbr 5143   E cep 5583   × cxp 5683  Ncnpi 10884   <N clti 10887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-xp 5691  df-lti 10915
This theorem is referenced by:  ltexpi  10942  ltapi  10943  ltmpi  10944  1lt2pi  10945  nlt1pi  10946  indpi  10947  nqereu  10969
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