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Theorem ordn2lp 6404
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 6402 . 2 (Ord 𝐴 → ¬ 𝐴𝐴)
2 ordtr 6398 . . 3 (Ord 𝐴 → Tr 𝐴)
3 trel 5268 . . 3 (Tr 𝐴 → ((𝐴𝐵𝐵𝐴) → 𝐴𝐴))
42, 3syl 17 . 2 (Ord 𝐴 → ((𝐴𝐵𝐵𝐴) → 𝐴𝐴))
51, 4mtod 198 1 (Ord 𝐴 → ¬ (𝐴𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2108  Tr wtr 5259  Ord word 6383
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-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-fr 5637  df-we 5639  df-ord 6387
This theorem is referenced by:  ordtri1  6417  ordnbtwn  6477  suc11  6491  smoord  8405  unblem1  9328  cantnfp1lem3  9720  cardprclem  10019  nosepssdm  27731  slerec  27864
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