Theorem ordelsuc 4197

Description: A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 29-Nov-2003.)

Assertion

Ref	Expression
ordelsuc	⊢ ((A ∈ 𝐶 ∧ Ord B) → (A ∈ B ↔ suc A ⊆ B))

Proof of Theorem ordelsuc

Step	Hyp	Ref	Expression
1		ordsucss 4196	. . 3 ⊢ (Ord B → (A ∈ B → suc A ⊆ B))
2	1	adantl 262	. 2 ⊢ ((A ∈ 𝐶 ∧ Ord B) → (A ∈ B → suc A ⊆ B))
3		sucssel 4127	. . 3 ⊢ (A ∈ 𝐶 → (suc A ⊆ B → A ∈ B))
4	3	adantr 261	. 2 ⊢ ((A ∈ 𝐶 ∧ Ord B) → (suc A ⊆ B → A ∈ B))
5	2, 4	impbid 120	1 ⊢ ((A ∈ 𝐶 ∧ Ord B) → (A ∈ B ↔ suc A ⊆ B))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1390   ⊆ wss 2911  Ord word 4065  suc csuc 4068

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-uni 3572  df-tr 3846  df-iord 4069  df-suc 4074

This theorem is referenced by:  onsucmin  4198  onsucelsucr  4199  onsucsssucr  4200  onsucsssucexmid  4212  ordgt0ge1  5957  nnsucsssuc  6010