Theorem onsucss 2359

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.

Hypotheses

Ref	Expression
on.1	⊢ A ∈ On
on.2	⊢ B ∈ On

Assertion

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

Proof of Theorem onsucss

Step	Hyp	Ref	Expression
1		on.1	. 2 ⊢ A ∈ On
2		on.2	. . 3 ⊢ B ∈ On
3	2	onord 2343	. 2 ⊢ Ord B
4		ordelsuc 2322	. 2 ⊢ ((A ∈ On ∧ Ord B) → (A ∈ B ↔ suc A ⊆ B))
5	1, 3, 4	mp2an 520	1 ⊢ (A ∈ B ↔ suc A ⊆ B)

Syntax hints:   ↔ wb 127   ∈ wcel 1092   ⊆ wss 1487  Ord word 2198  Oncon0 2199  suc csuc 2201

This theorem is referenced by:  limsuclem 2360  ranklon 3540

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205

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