Theorem ordelsuc 4177

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 4176	. . 3 ⊢ (Ord B → (A ∈ B → suc A ⊆ B))
2	1	adantl 262	. 2 ⊢ ((A ∈ 𝐶 ∧ Ord B) → (A ∈ B → suc A ⊆ B))
3		sucssel 4107	. . 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 1370   ⊆ wss 2890  Ord word 4044  suc csuc 4047

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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000

This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-sn 3352  df-uni 3551  df-tr 3825  df-iord 4048  df-suc 4053

This theorem is referenced by:  onsucmin  4178  onsucelsucr  4179  onsucsssucr  4180  onsucsssucexmid  4192  ordgt0ge1  5929  nnsucsssuc  5982