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Theorem ordnbtwn 6249
 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 6177 . . 3 (Ord 𝐴 → ¬ 𝐴𝐴)
2 ordn2lp 6179 . . . 4 (Ord 𝐴 → ¬ (𝐴𝐵𝐵𝐴))
3 pm2.24 124 . . . . 5 ((𝐴𝐵𝐵𝐴) → (¬ (𝐴𝐵𝐵𝐴) → 𝐴𝐴))
4 eleq2 2878 . . . . . . 7 (𝐵 = 𝐴 → (𝐴𝐵𝐴𝐴))
54biimpac 482 . . . . . 6 ((𝐴𝐵𝐵 = 𝐴) → 𝐴𝐴)
65a1d 25 . . . . 5 ((𝐴𝐵𝐵 = 𝐴) → (¬ (𝐴𝐵𝐵𝐴) → 𝐴𝐴))
73, 6jaodan 955 . . . 4 ((𝐴𝐵 ∧ (𝐵𝐴𝐵 = 𝐴)) → (¬ (𝐴𝐵𝐵𝐴) → 𝐴𝐴))
82, 7syl5com 31 . . 3 (Ord 𝐴 → ((𝐴𝐵 ∧ (𝐵𝐴𝐵 = 𝐴)) → 𝐴𝐴))
91, 8mtod 201 . 2 (Ord 𝐴 → ¬ (𝐴𝐵 ∧ (𝐵𝐴𝐵 = 𝐴)))
10 elsuci 6225 . . 3 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
1110anim2i 619 . 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 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  Ord word 6158  suc csuc 6161 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-fr 5478  df-we 5480  df-ord 6162  df-suc 6165 This theorem is referenced by:  onnbtwn  6250  ordsucss  7515
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