Theorem onsucb 4502

Description: A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc 4500. (Contributed by NM, 9-Sep-2003.)

Assertion

Ref	Expression
onsucb	⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)

Proof of Theorem onsucb

Step	Hyp	Ref	Expression
1		onsuc 4500	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
2		eloni 4375	. . 3 ⊢ (suc 𝐴 ∈ On → Ord suc 𝐴)
3		elex 2748	. . . . 5 ⊢ (suc 𝐴 ∈ On → suc 𝐴 ∈ V)
4		sucexb 4496	. . . . 5 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
5	3, 4	sylibr 134	. . . 4 ⊢ (suc 𝐴 ∈ On → 𝐴 ∈ V)
6		elong 4373	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
7		ordsucg 4501	. . . . 5 ⊢ (𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴))
8	6, 7	bitrd 188	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord suc 𝐴))
9	5, 8	syl 14	. . 3 ⊢ (suc 𝐴 ∈ On → (𝐴 ∈ On ↔ Ord suc 𝐴))
10	2, 9	mpbird 167	. 2 ⊢ (suc 𝐴 ∈ On → 𝐴 ∈ On)
11	1, 10	impbii 126	1 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)

Syntax hints:   ↔ wb 105   ∈ wcel 2148  Vcvv 2737  Ord word 4362  Oncon0 4363  suc csuc 4365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-tr 4102  df-iord 4366  df-on 4368  df-suc 4371

This theorem is referenced by:  onsucmin  4506  onsucuni2  4563