Theorem onsucssi 6933

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 5749	. 2 ⊢ Ord 𝐵
4		ordelsuc 6912	. 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 195   ∈ wcel 1977   ⊆ wss 3540  Ord word 5639  Oncon0 5640  suc csuc 5642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-suc 5646

This theorem is referenced by:  omopthlem1  7622  rankval4  8613  rankc1  8616  rankc2  8617  rankxplim  8625  rankxplim3  8627  onsucsuccmpi  31612