Theorem onsucss 3111

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 e. On
on.2	|- B e. On

Assertion

Ref	Expression
onsucss	|- (A e. B <-> suc A (_ B)

Proof of Theorem onsucss

Step	Hyp	Ref	Expression
1		on.1	. 2 |- A e. On
2		on.2	. . 3 |- B e. On
3	2	onord 3095	. 2 |- Ord B
4		ordelsuc 3071	. 2 |- ((A e. On /\ Ord B) -> (A e. B <-> suc A (_ B))
5	1, 3, 4	mp2an 697	1 |- (A e. B <-> suc A (_ B)

Syntax hints:   <-> wb 146   e. wcel 958   (_ wss 2047  Ord word 2947  Oncon0 2948  suc csuc 2950

This theorem is referenced by:  rankval4 4702  rankc1 4705  rankc2 4706  rankxplim 4712  rankxplim3 4714

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954

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