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Theorem ordn2lp 4269
 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 4267 . 2 (Ord 𝐴 → ¬ 𝐴𝐴)
2 ordtr 4115 . . 3 (Ord 𝐴 → Tr 𝐴)
3 trel 3861 . . 3 (Tr 𝐴 → ((𝐴𝐵𝐵𝐴) → 𝐴𝐴))
42, 3syl 14 . 2 (Ord 𝐴 → ((𝐴𝐵𝐵𝐴) → 𝐴𝐴))
51, 4mtod 589 1 (Ord 𝐴 → ¬ (𝐴𝐵𝐵𝐴))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∈ wcel 1393  Tr wtr 3854  Ord word 4099 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-sn 3381  df-uni 3581  df-tr 3855  df-iord 4103 This theorem is referenced by: (None)
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