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Theorem ordsssuc2 5731
 Description: An ordinal subset of an ordinal number belongs to its successor. (Contributed by NM, 1-Feb-2005.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
ordsssuc2 ((Ord 𝐴𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵))

Proof of Theorem ordsssuc2
StepHypRef Expression
1 elong 5648 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
21biimprd 237 . . . 4 (𝐴 ∈ V → (Ord 𝐴𝐴 ∈ On))
32anim1d 586 . . 3 (𝐴 ∈ V → ((Ord 𝐴𝐵 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On)))
4 onsssuc 5730 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵))
53, 4syl6 34 . 2 (𝐴 ∈ V → ((Ord 𝐴𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵)))
6 annim 440 . . . . 5 ((𝐵 ∈ On ∧ ¬ 𝐴 ∈ V) ↔ ¬ (𝐵 ∈ On → 𝐴 ∈ V))
7 ssexg 4732 . . . . . . 7 ((𝐴𝐵𝐵 ∈ On) → 𝐴 ∈ V)
87ex 449 . . . . . 6 (𝐴𝐵 → (𝐵 ∈ On → 𝐴 ∈ V))
9 elex 3185 . . . . . . 7 (𝐴 ∈ suc 𝐵𝐴 ∈ V)
109a1d 25 . . . . . 6 (𝐴 ∈ suc 𝐵 → (𝐵 ∈ On → 𝐴 ∈ V))
118, 10pm5.21ni 366 . . . . 5 (¬ (𝐵 ∈ On → 𝐴 ∈ V) → (𝐴𝐵𝐴 ∈ suc 𝐵))
126, 11sylbi 206 . . . 4 ((𝐵 ∈ On ∧ ¬ 𝐴 ∈ V) → (𝐴𝐵𝐴 ∈ suc 𝐵))
1312expcom 450 . . 3 𝐴 ∈ V → (𝐵 ∈ On → (𝐴𝐵𝐴 ∈ suc 𝐵)))
1413adantld 482 . 2 𝐴 ∈ V → ((Ord 𝐴𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵)))
155, 14pm2.61i 175 1 ((Ord 𝐴𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  Ord word 5639  Oncon0 5640  suc csuc 5642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-suc 5646 This theorem is referenced by:  ordunisuc2  6936
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