Theorem onsucssi 7792

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 6436	. 2 ⊢ Ord 𝐵
4		ordelsuc 7771	. 2 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵))
5	1, 3, 4	mp2an 693	1 ⊢ (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵)

Syntax hints:   ↔ wb 206   ∈ wcel 2114   ⊆ wss 3889  Ord word 6322  Oncon0 6323  suc csuc 6325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6326  df-on 6327  df-suc 6329

This theorem is referenced by:  omopthlem1  8595  rankval4  9791  rankc1  9794  rankc2  9795  rankxplim  9803  rankxplim3  9805  cuteq1  27809  bdayiun  27907  bdayn0p1  28361  rankval4b  35243  onsucsuccmpi  36625