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Theorem ordsucg 4194
 Description: The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.)
Assertion
Ref Expression
ordsucg (A V → (Ord A ↔ Ord suc A))

Proof of Theorem ordsucg
StepHypRef Expression
1 ordsucim 4192 . 2 (Ord A → Ord suc A)
2 sucidg 4119 . . 3 (A V → A suc A)
3 ordelord 4084 . . . 4 ((Ord suc A A suc A) → Ord A)
43ex 108 . . 3 (Ord suc A → (A suc A → Ord A))
52, 4syl5com 26 . 2 (A V → (Ord suc A → Ord A))
61, 5impbid2 131 1 (A V → (Ord A ↔ Ord suc A))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∈ wcel 1390  Vcvv 2551  Ord word 4065  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-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 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-sn 3373  df-uni 3572  df-tr 3846  df-iord 4069  df-suc 4074 This theorem is referenced by:  sucelon  4195
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