Theorem onsucb 4536

Description: A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc 4534. (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 4534	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
2		eloni 4407	. . 3 ⊢ (suc 𝐴 ∈ On → Ord suc 𝐴)
3		elex 2771	. . . . 5 ⊢ (suc 𝐴 ∈ On → suc 𝐴 ∈ V)
4		sucexb 4530	. . . . 5 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
5	3, 4	sylibr 134	. . . 4 ⊢ (suc 𝐴 ∈ On → 𝐴 ∈ V)
6		elong 4405	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
7		ordsucg 4535	. . . . 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 2164  Vcvv 2760  Ord word 4394  Oncon0 4395  suc csuc 4397

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-uni 3837  df-tr 4129  df-iord 4398  df-on 4400  df-suc 4403

This theorem is referenced by:  onsucmin  4540  onsucuni2  4597