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

Step	Hyp	Ref	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)
4	3	ex 108	. . 3 ⊢ (Ord suc A → (A ∈ suc A → Ord A))
5	2, 4	syl5com 26	. 2 ⊢ (A ∈ V → (Ord suc A → Ord A))
6	1, 5	impbid2 131	1 ⊢ (A ∈ V → (Ord A ↔ Ord suc A))

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