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Theorem ltpiord 7075
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 7063 . . 3 <N = ( E ∩ (N × N))
21breqi 3901 . 2 (𝐴 <N 𝐵𝐴( E ∩ (N × N))𝐵)
3 brinxp 4567 . . 3 ((𝐴N𝐵N) → (𝐴 E 𝐵𝐴( E ∩ (N × N))𝐵))
4 epelg 4172 . . . 4 (𝐵N → (𝐴 E 𝐵𝐴𝐵))
54adantl 273 . . 3 ((𝐴N𝐵N) → (𝐴 E 𝐵𝐴𝐵))
63, 5bitr3d 189 . 2 ((𝐴N𝐵N) → (𝐴( E ∩ (N × N))𝐵𝐴𝐵))
72, 6syl5bb 191 1 ((𝐴N𝐵N) → (𝐴 <N 𝐵𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 1463  cin 3036   class class class wbr 3895   E cep 4169   × cxp 4497  Ncnpi 7028   <N clti 7031
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-eprel 4171  df-xp 4505  df-lti 7063
This theorem is referenced by:  ltsopi  7076  pitric  7077  pitri3or  7078  ltdcpi  7079  ltexpi  7093  ltapig  7094  ltmpig  7095  1lt2pi  7096  nlt1pig  7097  archnqq  7173  prarloclemarch2  7175  prarloclemlt  7249  prarloclemn  7255
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