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Theorem ordsucg 4386
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 4384 . 2 (Ord 𝐴 → Ord suc 𝐴)
2 sucidg 4306 . . 3 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3 ordelord 4271 . . . 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 1463  Vcvv 2658  Ord word 4252  suc csuc 4255
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-uni 3705  df-tr 3995  df-iord 4256  df-suc 4261
This theorem is referenced by:  sucelon  4387
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