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Theorem ordsucg 4460
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 4458 . 2 (Ord 𝐴 → Ord suc 𝐴)
2 sucidg 4376 . . 3 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3 ordelord 4341 . . . 4 ((Ord suc 𝐴𝐴 ∈ suc 𝐴) → Ord 𝐴)
43ex 114 . . 3 (Ord suc 𝐴 → (𝐴 ∈ suc 𝐴 → Ord 𝐴))
52, 4syl5com 29 . 2 (𝐴 ∈ V → (Ord suc 𝐴 → Ord 𝐴))
61, 5impbid2 142 1 (𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 2128  Vcvv 2712  Ord word 4322  suc csuc 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-uni 3773  df-tr 4063  df-iord 4326  df-suc 4331
This theorem is referenced by:  sucelon  4461
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