Theorem onsuc 4502

Description: The successor of an ordinal number is an ordinal number. Closed form of onsuci 4517. Forward implication of onsucb 4504. 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 4377	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordsucim 4501	. . 3 ⊢ (Ord 𝐴 → Ord suc 𝐴)
3	1, 2	syl 14	. 2 ⊢ (𝐴 ∈ On → Ord suc 𝐴)
4		sucexg 4499	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V)
5		elong 4375	. . 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 2148  Vcvv 2739  Ord word 4364  Oncon0 4365  suc csuc 4367

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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  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 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370  df-suc 4373

This theorem is referenced by:  onsucb  4504  unon  4512  onsuci  4517  ordsucunielexmid  4532  tfrlemisucaccv  6328  tfrexlem  6337  tfri1dALT  6354  rdgisuc1  6387  rdgon  6389  oacl  6463  oasuc  6467  omsuc  6475