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Theorem ordnbtwn 6116
Description: There is no set between an ordinal class and its successor. Generalized Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 21-Jun-1998.) (Proof shortened by JJ, 24-Sep-2021.)
Assertion
Ref Expression
ordnbtwn (Ord 𝐴 → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))

Proof of Theorem ordnbtwn
StepHypRef Expression
1 ordirr 6044 . . 3 (Ord 𝐴 → ¬ 𝐴𝐴)
2 ordn2lp 6046 . . . 4 (Ord 𝐴 → ¬ (𝐴𝐵𝐵𝐴))
3 pm2.24 122 . . . . 5 ((𝐴𝐵𝐵𝐴) → (¬ (𝐴𝐵𝐵𝐴) → 𝐴𝐴))
4 eleq2 2847 . . . . . . 7 (𝐵 = 𝐴 → (𝐴𝐵𝐴𝐴))
54biimpac 471 . . . . . 6 ((𝐴𝐵𝐵 = 𝐴) → 𝐴𝐴)
65a1d 25 . . . . 5 ((𝐴𝐵𝐵 = 𝐴) → (¬ (𝐴𝐵𝐵𝐴) → 𝐴𝐴))
73, 6jaodan 941 . . . 4 ((𝐴𝐵 ∧ (𝐵𝐴𝐵 = 𝐴)) → (¬ (𝐴𝐵𝐵𝐴) → 𝐴𝐴))
82, 7syl5com 31 . . 3 (Ord 𝐴 → ((𝐴𝐵 ∧ (𝐵𝐴𝐵 = 𝐴)) → 𝐴𝐴))
91, 8mtod 190 . 2 (Ord 𝐴 → ¬ (𝐴𝐵 ∧ (𝐵𝐴𝐵 = 𝐴)))
10 elsuci 6092 . . 3 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
1110anim2i 608 . 2 ((𝐴𝐵𝐵 ∈ suc 𝐴) → (𝐴𝐵 ∧ (𝐵𝐴𝐵 = 𝐴)))
129, 11nsyl 138 1 (Ord 𝐴 → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387  wo 834   = wceq 1508  wcel 2051  Ord word 6025  suc csuc 6028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-tr 5027  df-eprel 5313  df-fr 5362  df-we 5364  df-ord 6029  df-suc 6032
This theorem is referenced by:  onnbtwn  6117  ordsucss  7347
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