Theorem onsuc 4534

Description: The successor of an ordinal number is an ordinal number. Closed form of onsuci 4549. Forward implication of onsucb 4536. 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 4407	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordsucim 4533	. . 3 ⊢ (Ord 𝐴 → Ord suc 𝐴)
3	1, 2	syl 14	. 2 ⊢ (𝐴 ∈ On → Ord suc 𝐴)
4		sucexg 4531	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V)
5		elong 4405	. . 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 2164  Vcvv 2760  Ord word 4394  Oncon0 4395  suc csuc 4397

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-uni 3837  df-tr 4129  df-iord 4398  df-on 4400  df-suc 4403

This theorem is referenced by:  onsucb  4536  unon  4544  onsuci  4549  ordsucunielexmid  4564  tfrlemisucaccv  6380  tfrexlem  6389  tfri1dALT  6406  rdgisuc1  6439  rdgon  6441  oacl  6515  oasuc  6519  omsuc  6527