Theorem onmindif 2312

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 2232	. . . . . . . . . . 11 |- ((x e. On /\ B e. On) -> (x (_ B <-> -. B e. x))
2		onsssuc 2311	. . . . . . . . . . 11 |- ((x e. On /\ B e. On) -> (x (_ B <-> x e. suc B))
3	1, 2	bitr3d 408	. . . . . . . . . 10 |- ((x e. On /\ B e. On) -> (-. B e. x <-> x e. suc B))
4	3	bicon1d 405	. . . . . . . . 9 |- ((x e. On /\ B e. On) -> (-. x e. suc B <-> B e. x))
5		ssel2 1503	. . . . . . . . 9 |- ((A (_ On /\ x e. A) -> x e. On)
6	4, 5	sylan 343	. . . . . . . 8 |- (((A (_ On /\ x e. A) /\ B e. On) -> (-. x e. suc B <-> B e. x))
7	6	biimpd 135	. . . . . . 7 |- (((A (_ On /\ x e. A) /\ B e. On) -> (-. x e. suc B -> B e. x))
8	7	exp31 293	. . . . . 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 283	. . . 4 |- ((A (_ On /\ B e. On) -> ((x e. A /\ -. x e. suc B) -> B e. x))
11		eldif 1496	. . . 4 |- (x e. (A \ suc B) <-> (x e. A /\ -. x e. suc B))
12	10, 11	syl5ib 181	. . 3 |- ((A (_ On /\ B e. On) -> (x e. (A \ suc B) -> B e. x))
13	12	r19.21aiv 1259	. 2 |- ((A (_ On /\ B e. On) -> A.x e. (A \ suc B)B e. x)
14		elintg 1973	. . 3 |- (B e. On -> (B e. |^|(A \ suc B) <-> A.x e. (A \ suc B)B e. x))
15	14	adantl 305	. 2 |- ((A (_ On /\ B e. On) -> (B e. |^|(A \ suc B) <-> A.x e. (A \ suc B)B e. x))
16	13, 15	mpbird 171	1 |- ((A (_ On /\ B e. On) -> B e. |^|(A \ suc B))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   e. wcel 1092  A.wral 1201   \ cdif 1484   (_ wss 1487  |^|cint 1965  Oncon0 2199  suc csuc 2201

This theorem is referenced by:  unblem3 3433

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205

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