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Theorem ordsssuc2 5956
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 5873 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
21biimprd 238 . . . 4 (𝐴 ∈ V → (Ord 𝐴𝐴 ∈ On))
32anim1d 598 . . 3 (𝐴 ∈ V → ((Ord 𝐴𝐵 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On)))
4 onsssuc 5955 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵))
53, 4syl6 35 . 2 (𝐴 ∈ V → ((Ord 𝐴𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵)))
6 annim 390 . . . . 5 ((𝐵 ∈ On ∧ ¬ 𝐴 ∈ V) ↔ ¬ (𝐵 ∈ On → 𝐴 ∈ V))
7 ssexg 4939 . . . . . . 7 ((𝐴𝐵𝐵 ∈ On) → 𝐴 ∈ V)
87ex 397 . . . . . 6 (𝐴𝐵 → (𝐵 ∈ On → 𝐴 ∈ V))
9 elex 3364 . . . . . . 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 207 . . . 4 ((𝐵 ∈ On ∧ ¬ 𝐴 ∈ V) → (𝐴𝐵𝐴 ∈ suc 𝐵))
1312expcom 398 . . 3 𝐴 ∈ V → (𝐵 ∈ On → (𝐴𝐵𝐴 ∈ suc 𝐵)))
1413adantld 478 . 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 382  wcel 2145  Vcvv 3351  wss 3723  Ord word 5864  Oncon0 5865  suc csuc 5867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-tr 4888  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-ord 5868  df-on 5869  df-suc 5871
This theorem is referenced by:  ordunisuc2  7195
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