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Theorem ordsssuc2 6027
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 5944 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
21biimprd 239 . . . 4 (𝐴 ∈ V → (Ord 𝐴𝐴 ∈ On))
32anim1d 600 . . 3 (𝐴 ∈ V → ((Ord 𝐴𝐵 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On)))
4 onsssuc 6026 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵))
53, 4syl6 35 . 2 (𝐴 ∈ V → ((Ord 𝐴𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵)))
6 annim 392 . . . . 5 ((𝐵 ∈ On ∧ ¬ 𝐴 ∈ V) ↔ ¬ (𝐵 ∈ On → 𝐴 ∈ V))
7 ssexg 4999 . . . . . . 7 ((𝐴𝐵𝐵 ∈ On) → 𝐴 ∈ V)
87ex 399 . . . . . 6 (𝐴𝐵 → (𝐵 ∈ On → 𝐴 ∈ V))
9 elex 3406 . . . . . . 7 (𝐴 ∈ suc 𝐵𝐴 ∈ V)
109a1d 25 . . . . . 6 (𝐴 ∈ suc 𝐵 → (𝐵 ∈ On → 𝐴 ∈ V))
118, 10pm5.21ni 368 . . . . 5 (¬ (𝐵 ∈ On → 𝐴 ∈ V) → (𝐴𝐵𝐴 ∈ suc 𝐵))
126, 11sylbi 208 . . . 4 ((𝐵 ∈ On ∧ ¬ 𝐴 ∈ V) → (𝐴𝐵𝐴 ∈ suc 𝐵))
1312expcom 400 . . 3 𝐴 ∈ V → (𝐵 ∈ On → (𝐴𝐵𝐴 ∈ suc 𝐵)))
1413adantld 480 . 2 𝐴 ∈ V → ((Ord 𝐴𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵)))
155, 14pm2.61i 176 1 ((Ord 𝐴𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wcel 2156  Vcvv 3391  wss 3769  Ord word 5935  Oncon0 5936  suc csuc 5938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-tr 4947  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-ord 5939  df-on 5940  df-suc 5942
This theorem is referenced by:  ordunisuc2  7274
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