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Theorem sucelon 4195
Description: The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)
Assertion
Ref Expression
sucelon (A On ↔ suc A On)

Proof of Theorem sucelon
StepHypRef Expression
1 suceloni 4193 . 2 (A On → suc A On)
2 eloni 4078 . . 3 (suc A On → Ord suc A)
3 elex 2560 . . . . 5 (suc A On → suc A V)
4 sucexb 4189 . . . . 5 (A V ↔ suc A V)
53, 4sylibr 137 . . . 4 (suc A On → A V)
6 elong 4076 . . . . 5 (A V → (A On ↔ Ord A))
7 ordsucg 4194 . . . . 5 (A V → (Ord A ↔ Ord suc A))
86, 7bitrd 177 . . . 4 (A V → (A On ↔ Ord suc A))
95, 8syl 14 . . 3 (suc A On → (A On ↔ Ord suc A))
102, 9mpbird 156 . 2 (suc A On → A On)
111, 10impbii 117 1 (A On ↔ suc A On)
Colors of variables: wff set class
Syntax hints:  wb 98   wcel 1390  Vcvv 2551  Ord word 4065  Oncon0 4066  suc csuc 4068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074
This theorem is referenced by:  onsucmin  4198
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