Theorem ordelsuc 7642

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 7640	. . 3 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
2	1	adantl 481	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
3		sucssel 6343	. . 3 ⊢ (𝐴 ∈ 𝐶 → (suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵))
4	3	adantr 480	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ Ord 𝐵) → (suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵))
5	2, 4	impbid 211	1 ⊢ ((𝐴 ∈ 𝐶 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2108   ⊆ wss 3883  Ord word 6250  suc csuc 6253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255  df-suc 6257

This theorem is referenced by:  onsucmin  7643  onsucssi  7663  tfindsg2  7683  ordgt0ge1  8289  onomeneq  8943  cantnflem1  9377  r1ordg  9467  r1val1  9475  rankonidlem  9517  rankxplim3  9570  ttrcltr  33702  dmttrcl  33707