Theorem onsucb 4630

Description: A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc 4628. (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 4628	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
2		eloni 4501	. . 3 ⊢ (suc 𝐴 ∈ On → Ord suc 𝐴)
3		elex 2827	. . . . 5 ⊢ (suc 𝐴 ∈ On → suc 𝐴 ∈ V)
4		sucexb 4624	. . . . 5 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
5	3, 4	sylibr 134	. . . 4 ⊢ (suc 𝐴 ∈ On → 𝐴 ∈ V)
6		elong 4499	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
7		ordsucg 4629	. . . . 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 2205  Vcvv 2815  Ord word 4488  Oncon0 4489  suc csuc 4491

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-tr 4214  df-iord 4492  df-on 4494  df-suc 4497

This theorem is referenced by:  onsucmin  4634  onsucuni2  4691