Theorem onsuc 4592

Description: The successor of an ordinal number is an ordinal number. Closed form of onsuci 4607. Forward implication of onsucb 4594. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)

Assertion

Ref	Expression
onsuc	⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)

Proof of Theorem onsuc

Step	Hyp	Ref	Expression
1		eloni 4465	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordsucim 4591	. . 3 ⊢ (Ord 𝐴 → Ord suc 𝐴)
3	1, 2	syl 14	. 2 ⊢ (𝐴 ∈ On → Ord suc 𝐴)
4		sucexg 4589	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V)
5		elong 4463	. . 3 ⊢ (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
6	4, 5	syl 14	. 2 ⊢ (𝐴 ∈ On → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
7	3, 6	mpbird 167	1 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)

Syntax hints:   → wi 4   ↔ wb 105   ∈ wcel 2200  Vcvv 2799  Ord word 4452  Oncon0 4453  suc csuc 4455

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3888  df-tr 4182  df-iord 4456  df-on 4458  df-suc 4461

This theorem is referenced by:  onsucb  4594  unon  4602  onsuci  4607  ordsucunielexmid  4622  tfrlemisucaccv  6469  tfrexlem  6478  tfri1dALT  6495  rdgisuc1  6528  rdgon  6530  oacl  6604  oasuc  6608  omsuc  6616