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Theorem ordn2lp 5781
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 5779 . 2 (Ord 𝐴 → ¬ 𝐴𝐴)
2 ordtr 5775 . . 3 (Ord 𝐴 → Tr 𝐴)
3 trel 4792 . . 3 (Tr 𝐴 → ((𝐴𝐵𝐵𝐴) → 𝐴𝐴))
42, 3syl 17 . 2 (Ord 𝐴 → ((𝐴𝐵𝐵𝐴) → 𝐴𝐴))
51, 4mtod 189 1 (Ord 𝐴 → ¬ (𝐴𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wcel 2030  Tr wtr 4785  Ord word 5760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-fr 5102  df-we 5104  df-ord 5764
This theorem is referenced by:  ordtri1  5794  ordnbtwn  5854  ordnbtwnOLD  5855  suc11  5869  smoord  7507  unblem1  8253  cantnfp1lem3  8615  cardprclem  8843  nosepssdm  31961  slerec  32048
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