Theorem onsuc 4567

Description: The successor of an ordinal number is an ordinal number. Closed form of onsuci 4582. Forward implication of onsucb 4569. 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 4440	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordsucim 4566	. . 3 ⊢ (Ord 𝐴 → Ord suc 𝐴)
3	1, 2	syl 14	. 2 ⊢ (𝐴 ∈ On → Ord suc 𝐴)
4		sucexg 4564	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V)
5		elong 4438	. . 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 2178  Vcvv 2776  Ord word 4427  Oncon0 4428  suc csuc 4430

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 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-uni 3865  df-tr 4159  df-iord 4431  df-on 4433  df-suc 4436

This theorem is referenced by:  onsucb  4569  unon  4577  onsuci  4582  ordsucunielexmid  4597  tfrlemisucaccv  6434  tfrexlem  6443  tfri1dALT  6460  rdgisuc1  6493  rdgon  6495  oacl  6569  oasuc  6573  omsuc  6581