Theorem ordelsuc 4249

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 4248	. . 3 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
2	1	adantl 271	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
3		sucssel 4179	. . 3 ⊢ (𝐴 ∈ 𝐶 → (suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵))
4	3	adantr 270	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ Ord 𝐵) → (suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵))
5	2, 4	impbid 127	1 ⊢ ((𝐴 ∈ 𝐶 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∈ wcel 1433   ⊆ wss 2973  Ord word 4117  suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-uni 3602  df-tr 3876  df-iord 4121  df-suc 4126

This theorem is referenced by:  onsucssi  4250  onsucmin  4251  onsucelsucr  4252  onsucsssucr  4253  onsucsssucexmid  4270  ordgt0ge1  6041  nnsucsssuc  6094