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Theorem ordsssuc2 5773
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 5690 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
21biimprd 238 . . . 4 (𝐴 ∈ V → (Ord 𝐴𝐴 ∈ On))
32anim1d 587 . . 3 (𝐴 ∈ V → ((Ord 𝐴𝐵 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On)))
4 onsssuc 5772 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵))
53, 4syl6 35 . 2 (𝐴 ∈ V → ((Ord 𝐴𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵)))
6 annim 441 . . . . 5 ((𝐵 ∈ On ∧ ¬ 𝐴 ∈ V) ↔ ¬ (𝐵 ∈ On → 𝐴 ∈ V))
7 ssexg 4764 . . . . . . 7 ((𝐴𝐵𝐵 ∈ On) → 𝐴 ∈ V)
87ex 450 . . . . . 6 (𝐴𝐵 → (𝐵 ∈ On → 𝐴 ∈ V))
9 elex 3198 . . . . . . 7 (𝐴 ∈ suc 𝐵𝐴 ∈ V)
109a1d 25 . . . . . 6 (𝐴 ∈ suc 𝐵 → (𝐵 ∈ On → 𝐴 ∈ V))
118, 10pm5.21ni 367 . . . . 5 (¬ (𝐵 ∈ On → 𝐴 ∈ V) → (𝐴𝐵𝐴 ∈ suc 𝐵))
126, 11sylbi 207 . . . 4 ((𝐵 ∈ On ∧ ¬ 𝐴 ∈ V) → (𝐴𝐵𝐴 ∈ suc 𝐵))
1312expcom 451 . . 3 𝐴 ∈ V → (𝐵 ∈ On → (𝐴𝐵𝐴 ∈ suc 𝐵)))
1413adantld 483 . 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 196  wa 384  wcel 1987  Vcvv 3186  wss 3555  Ord word 5681  Oncon0 5682  suc csuc 5684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685  df-on 5686  df-suc 5688
This theorem is referenced by:  ordunisuc2  6991
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