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Theorem ordsucg 4256
 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 4254 . 2 (Ord 𝐴 → Ord suc 𝐴)
2 sucidg 4181 . . 3 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3 ordelord 4146 . . . 4 ((Ord suc 𝐴𝐴 ∈ suc 𝐴) → Ord 𝐴)
43ex 112 . . 3 (Ord suc 𝐴 → (𝐴 ∈ suc 𝐴 → Ord 𝐴))
52, 4syl5com 29 . 2 (𝐴 ∈ V → (Ord suc 𝐴 → Ord 𝐴))
61, 5impbid2 135 1 (𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102   ∈ wcel 1409  Vcvv 2574  Ord word 4127  suc csuc 4130 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-uni 3609  df-tr 3883  df-iord 4131  df-suc 4136 This theorem is referenced by:  sucelon  4257
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