Theorem onsucssi 7837

Description: A set belongs to an ordinal number iff its successor is a subset of the ordinal number. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 16-Sep-1995.)

Hypotheses

Ref	Expression
onssi.1	⊢ 𝐴 ∈ On
onsucssi.2	⊢ 𝐵 ∈ On

Assertion

Ref	Expression
onsucssi	⊢ (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵)

Proof of Theorem onsucssi

Step	Hyp	Ref	Expression
1		onssi.1	. 2 ⊢ 𝐴 ∈ On
2		onsucssi.2	. . 3 ⊢ 𝐵 ∈ On
3	2	onordi 6475	. 2 ⊢ Ord 𝐵
4		ordelsuc 7816	. 2 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵))
5	1, 3, 4	mp2an 704	1 ⊢ (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵)

Syntax hints:   ↔ wb 209   ∈ wcel 2149   ⊆ wss 3913  Ord word 6360  Oncon0 6361  suc csuc 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-tr 5223  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-ord 6364  df-on 6365  df-suc 6367

This theorem is referenced by:  omopthlem1  8645  rankval4  9839  rankc1  9842  rankc2  9843  rankxplim  9851  rankxplim3  9853  cuteq1  27976  bdayiun  28074  bdayn0p1  28528  rankval4b  35436  onsucsuccmpi  36843