Theorem onsucb 4559

Description: A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc 4557. (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 4557	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
2		eloni 4430	. . 3 ⊢ (suc 𝐴 ∈ On → Ord suc 𝐴)
3		elex 2785	. . . . 5 ⊢ (suc 𝐴 ∈ On → suc 𝐴 ∈ V)
4		sucexb 4553	. . . . 5 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
5	3, 4	sylibr 134	. . . 4 ⊢ (suc 𝐴 ∈ On → 𝐴 ∈ V)
6		elong 4428	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
7		ordsucg 4558	. . . . 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 2177  Vcvv 2773  Ord word 4417  Oncon0 4418  suc csuc 4420

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-uni 3857  df-tr 4151  df-iord 4421  df-on 4423  df-suc 4426

This theorem is referenced by:  onsucmin  4563  onsucuni2  4620