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Theorem ordsssuc2 6161
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 6081 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
21biimprd 249 . . . 4 (𝐴 ∈ V → (Ord 𝐴𝐴 ∈ On))
32anim1d 610 . . 3 (𝐴 ∈ V → ((Ord 𝐴𝐵 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On)))
4 onsssuc 6160 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵))
53, 4syl6 35 . 2 (𝐴 ∈ V → ((Ord 𝐴𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵)))
6 annim 404 . . . . 5 ((𝐵 ∈ On ∧ ¬ 𝐴 ∈ V) ↔ ¬ (𝐵 ∈ On → 𝐴 ∈ V))
7 ssexg 5125 . . . . . . 7 ((𝐴𝐵𝐵 ∈ On) → 𝐴 ∈ V)
87ex 413 . . . . . 6 (𝐴𝐵 → (𝐵 ∈ On → 𝐴 ∈ V))
9 elex 3458 . . . . . . 7 (𝐴 ∈ suc 𝐵𝐴 ∈ V)
109a1d 25 . . . . . 6 (𝐴 ∈ suc 𝐵 → (𝐵 ∈ On → 𝐴 ∈ V))
118, 10pm5.21ni 379 . . . . 5 (¬ (𝐵 ∈ On → 𝐴 ∈ V) → (𝐴𝐵𝐴 ∈ suc 𝐵))
126, 11sylbi 218 . . . 4 ((𝐵 ∈ On ∧ ¬ 𝐴 ∈ V) → (𝐴𝐵𝐴 ∈ suc 𝐵))
1312expcom 414 . . 3 𝐴 ∈ V → (𝐵 ∈ On → (𝐴𝐵𝐴 ∈ suc 𝐵)))
1413adantld 491 . 2 𝐴 ∈ V → ((Ord 𝐴𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵)))
155, 14pm2.61i 183 1 ((Ord 𝐴𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wcel 2083  Vcvv 3440  wss 3865  Ord word 6072  Oncon0 6073  suc csuc 6075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-tr 5071  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-ord 6076  df-on 6077  df-suc 6079
This theorem is referenced by:  ordunisuc2  7422
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