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Theorem ordn2lp 4455
Description: An ordinal class cannot be an element of one of its members. Variant of first part of Theorem 2.2(vii) of [BellMachover] p. 469. (Contributed by NM, 3-Apr-1994.)
Assertion
Ref Expression
ordn2lp (Ord 𝐴 → ¬ (𝐴𝐵𝐵𝐴))

Proof of Theorem ordn2lp
StepHypRef Expression
1 ordirr 4452 . 2 (Ord 𝐴 → ¬ 𝐴𝐴)
2 ordtr 4295 . . 3 (Ord 𝐴 → Tr 𝐴)
3 trel 4028 . . 3 (Tr 𝐴 → ((𝐴𝐵𝐵𝐴) → 𝐴𝐴))
42, 3syl 14 . 2 (Ord 𝐴 → ((𝐴𝐵𝐵𝐴) → 𝐴𝐴))
51, 4mtod 652 1 (Ord 𝐴 → ¬ (𝐴𝐵𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wcel 1480  Tr wtr 4021  Ord word 4279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-setind 4447
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079  df-sn 3528  df-uni 3732  df-tr 4022  df-iord 4283
This theorem is referenced by:  nninfalllemn  13191
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