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Theorem ordnbtwn 6274
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 6202 . . 3 (Ord 𝐴 → ¬ 𝐴𝐴)
2 ordn2lp 6204 . . . 4 (Ord 𝐴 → ¬ (𝐴𝐵𝐵𝐴))
3 pm2.24 124 . . . . 5 ((𝐴𝐵𝐵𝐴) → (¬ (𝐴𝐵𝐵𝐴) → 𝐴𝐴))
4 eleq2 2898 . . . . . . 7 (𝐵 = 𝐴 → (𝐴𝐵𝐴𝐴))
54biimpac 479 . . . . . 6 ((𝐴𝐵𝐵 = 𝐴) → 𝐴𝐴)
65a1d 25 . . . . 5 ((𝐴𝐵𝐵 = 𝐴) → (¬ (𝐴𝐵𝐵𝐴) → 𝐴𝐴))
73, 6jaodan 951 . . . 4 ((𝐴𝐵 ∧ (𝐵𝐴𝐵 = 𝐴)) → (¬ (𝐴𝐵𝐵𝐴) → 𝐴𝐴))
82, 7syl5com 31 . . 3 (Ord 𝐴 → ((𝐴𝐵 ∧ (𝐵𝐴𝐵 = 𝐴)) → 𝐴𝐴))
91, 8mtod 199 . 2 (Ord 𝐴 → ¬ (𝐴𝐵 ∧ (𝐵𝐴𝐵 = 𝐴)))
10 elsuci 6250 . . 3 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
1110anim2i 616 . 2 ((𝐴𝐵𝐵 ∈ suc 𝐴) → (𝐴𝐵 ∧ (𝐵𝐴𝐵 = 𝐴)))
129, 11nsyl 142 1 (Ord 𝐴 → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 841   = wceq 1528  wcel 2105  Ord word 6183  suc csuc 6186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-fr 5507  df-we 5509  df-ord 6187  df-suc 6190
This theorem is referenced by:  onnbtwn  6275  ordsucss  7522
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