Theorem onsucssi 4542

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
onsucssi.1	|- A e. On
onsucssi.2	|- B e. On

Assertion

Ref	Expression
onsucssi	|- ( A e. B <-> suc A C_ B )

Proof of Theorem onsucssi

Step	Hyp	Ref	Expression
1		onsucssi.1	. 2 |- A e. On
2		onsucssi.2	. . 3 |- B e. On
3	2	onordi 4461	. 2 |- Ord B
4		ordelsuc 4541	. 2 |- ( ( A e. On /\ Ord B ) -> ( A e. B <-> suc A C_ B ) )
5	1, 3, 4	mp2an 426	1 |- ( A e. B <-> suc A C_ B )

Syntax hints:   <-> wb 105   e. wcel 2167   C_ wss 3157   Ord word 4397   Oncon0 4398   suc csuc 4400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-uni 3840  df-tr 4132  df-iord 4401  df-on 4403  df-suc 4406

This theorem is referenced by: (None)