Theorem onmindif 3050

Description: When its successor is subtracted from a class of ordinal numbers, an ordinal number is less than the minimum of the resulting subclass.

Assertion

Ref	Expression
onmindif	|- ((A (_ On /\ B e. On) -> B e. |^|(A \ suc B))

Proof of Theorem onmindif

Step	Hyp	Ref	Expression
1		ontri1 2971	. . . . . . . . . . 11 |- ((x e. On /\ B e. On) -> (x (_ B <-> -. B e. x))
2		onsssuc 3048	. . . . . . . . . . 11 |- ((x e. On /\ B e. On) -> (x (_ B <-> x e. suc B))
3	1, 2	bitr3d 528	. . . . . . . . . 10 |- ((x e. On /\ B e. On) -> (-. B e. x <-> x e. suc B))
4	3	con1bid 525	. . . . . . . . 9 |- ((x e. On /\ B e. On) -> (-. x e. suc B <-> B e. x))
5		ssel2 2054	. . . . . . . . 9 |- ((A (_ On /\ x e. A) -> x e. On)
6	4, 5	sylan 448	. . . . . . . 8 |- (((A (_ On /\ x e. A) /\ B e. On) -> (-. x e. suc B <-> B e. x))
7	6	biimpd 153	. . . . . . 7 |- (((A (_ On /\ x e. A) /\ B e. On) -> (-. x e. suc B -> B e. x))
8	7	exp31 376	. . . . . 6 |- (A (_ On -> (x e. A -> (B e. On -> (-. x e. suc B -> B e. x))))
9	8	com23 32	. . . . 5 |- (A (_ On -> (B e. On -> (x e. A -> (-. x e. suc B -> B e. x))))
10	9	imp4b 365	. . . 4 |- ((A (_ On /\ B e. On) -> ((x e. A /\ -. x e. suc B) -> B e. x))
11		eldif 2047	. . . 4 |- (x e. (A \ suc B) <-> (x e. A /\ -. x e. suc B))
12	10, 11	syl5ib 206	. . 3 |- ((A (_ On /\ B e. On) -> (x e. (A \ suc B) -> B e. x))
13	12	r19.21aiv 1705	. 2 |- ((A (_ On /\ B e. On) -> A.x e. (A \ suc B)B e. x)
14		elintg 2531	. . 3 |- (B e. On -> (B e. |^|(A \ suc B) <-> A.x e. (A \ suc B)B e. x))
15	14	adantl 388	. 2 |- ((A (_ On /\ B e. On) -> (B e. |^|(A \ suc B) <-> A.x e. (A \ suc B)B e. x))
16	13, 15	mpbird 196	1 |- ((A (_ On /\ B e. On) -> B e. |^|(A \ suc B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 955  A.wral 1637   \ cdif 2034   (_ wss 2037  |^|cint 2523  Oncon0 2938  suc csuc 2940

This theorem is referenced by:  unblem3 4519

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-int 2524  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944

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