Theorem onsucss 3107

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 3091	. 2 |- Ord B
4		ordelsuc 3067	. 2 |- ((A e. On /\ Ord B) -> (A e. B <-> suc A (_ B))
5	1, 3, 4	mp2an 696	1 |- (A e. B <-> suc A (_ B)

Syntax hints:   <-> wb 146   e. wcel 957   (_ wss 2044  Ord word 2943  Oncon0 2944  suc csuc 2946

This theorem is referenced by:  rankval4 4685  rankc1 4688  rankc2 4689  rankxplim 4695  rankxplim3 4697

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-suc 2950

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