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Theorem ordsucg 4309
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 4307 . 2 (Ord 𝐴 → Ord suc 𝐴)
2 sucidg 4234 . . 3 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3 ordelord 4199 . . . 4 ((Ord suc 𝐴𝐴 ∈ suc 𝐴) → Ord 𝐴)
43ex 113 . . 3 (Ord suc 𝐴 → (𝐴 ∈ suc 𝐴 → Ord 𝐴))
52, 4syl5com 29 . 2 (𝐴 ∈ V → (Ord suc 𝐴 → Ord 𝐴))
61, 5impbid2 141 1 (𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wcel 1438  Vcvv 2619  Ord word 4180  suc csuc 4183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-uni 3649  df-tr 3929  df-iord 4184  df-suc 4189
This theorem is referenced by:  sucelon  4310
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