Theorem ordsssuc2 3065

Description: An ordinal subset of an ordinal number belongs to its successor.

Assertion

Ref	Expression
ordsssuc2	|- ((Ord A /\ B e. On) -> (A (_ B <-> A e. suc B))

Proof of Theorem ordsssuc2

Step	Hyp	Ref	Expression
1		ssexg 2726	. . . . . . 7 |- ((A (_ B /\ B e. On) -> A e. V)
2		elong 2962	. . . . . . 7 |- (A e. V -> (A e. On <-> Ord A))
3	1, 2	syl 10	. . . . . 6 |- ((A (_ B /\ B e. On) -> (A e. On <-> Ord A))
4		onsssuc 3064	. . . . . . . . . 10 |- ((A e. On /\ B e. On) -> (A (_ B <-> A e. suc B))
5	4	biimpd 153	. . . . . . . . 9 |- ((A e. On /\ B e. On) -> (A (_ B -> A e. suc B))
6	5	ex 373	. . . . . . . 8 |- (A e. On -> (B e. On -> (A (_ B -> A e. suc B)))
7	6	com13 33	. . . . . . 7 |- (A (_ B -> (B e. On -> (A e. On -> A e. suc B)))
8	7	imp 350	. . . . . 6 |- ((A (_ B /\ B e. On) -> (A e. On -> A e. suc B))
9	3, 8	sylbird 205	. . . . 5 |- ((A (_ B /\ B e. On) -> (Ord A -> A e. suc B))
10	9	ex 373	. . . 4 |- (A (_ B -> (B e. On -> (Ord A -> A e. suc B)))
11	10	com13 33	. . 3 |- (Ord A -> (B e. On -> (A (_ B -> A e. suc B)))
12	11	imp 350	. 2 |- ((Ord A /\ B e. On) -> (A (_ B -> A e. suc B))
13		elong 2962	. . . . 5 |- (A e. suc B -> (A e. On <-> Ord A))
14	4	biimprd 154	. . . . . . 7 |- ((A e. On /\ B e. On) -> (A e. suc B -> A (_ B))
15	14	ex 373	. . . . . 6 |- (A e. On -> (B e. On -> (A e. suc B -> A (_ B)))
16	15	com3r 35	. . . . 5 |- (A e. suc B -> (A e. On -> (B e. On -> A (_ B)))
17	13, 16	sylbird 205	. . . 4 |- (A e. suc B -> (Ord A -> (B e. On -> A (_ B)))
18	17	com3l 34	. . 3 |- (Ord A -> (B e. On -> (A e. suc B -> A (_ B)))
19	18	imp 350	. 2 |- ((Ord A /\ B e. On) -> (A e. suc B -> A (_ B))
20	12, 19	impbid 518	1 |- ((Ord A /\ B e. On) -> (A (_ B <-> A e. suc B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 960  Vcvv 1814   (_ wss 2050  Ord word 2953  Oncon0 2954  suc csuc 2956

This theorem is referenced by:  ordunisuc2 3121

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-suc 2960

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