Theorem onmindif 3760

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 C_ On /\ B e. On) -> B e. |^|(A \ suc B))

Proof of Theorem onmindif

Step	Hyp	Ref	Expression
1		ontri1 3695	. . . . . . . . . . 11 |- ((x e. On /\ B e. On) -> (x C_ B <-> -. B e. x))
2		onsssuc 3757	. . . . . . . . . . 11 |- ((x e. On /\ B e. On) -> (x C_ B <-> x e. suc B))
3	1, 2	bitr3d 589	. . . . . . . . . 10 |- ((x e. On /\ B e. On) -> (-. B e. x <-> x e. suc B))
4	3	con1bid 586	. . . . . . . . 9 |- ((x e. On /\ B e. On) -> (-. x e. suc B <-> B e. x))
5		ssel2 2616	. . . . . . . . 9 |- ((A C_ On /\ x e. A) -> x e. On)
6	4, 5	sylan 497	. . . . . . . 8 |- (((A C_ On /\ x e. A) /\ B e. On) -> (-. x e. suc B <-> B e. x))
7	6	biimpd 170	. . . . . . 7 |- (((A C_ On /\ x e. A) /\ B e. On) -> (-. x e. suc B -> B e. x))
8	7	exp31 407	. . . . . 6 |- (A C_ On -> (x e. A -> (B e. On -> (-. x e. suc B -> B e. x))))
9	8	com23 36	. . . . 5 |- (A C_ On -> (B e. On -> (x e. A -> (-. x e. suc B -> B e. x))))
10	9	imp4b 392	. . . 4 |- ((A C_ On /\ B e. On) -> ((x e. A /\ -. x e. suc B) -> B e. x))
11		eldif 2609	. . . 4 |- (x e. (A \ suc B) <-> (x e. A /\ -. x e. suc B))
12	10, 11	syl5ib 223	. . 3 |- ((A C_ On /\ B e. On) -> (x e. (A \ suc B) -> B e. x))
13	12	r19.21aiv 2175	. 2 |- ((A C_ On /\ B e. On) -> A.x e. (A \ suc B)B e. x)
14		elintg 3222	. . 3 |- (B e. On -> (B e. |^|(A \ suc B) <-> A.x e. (A \ suc B)B e. x))
15	14	adantl 424	. 2 |- ((A C_ On /\ B e. On) -> (B e. |^|(A \ suc B) <-> A.x e. (A \ suc B)B e. x))
16	13, 15	mpbird 213	1 |- ((A C_ On /\ B e. On) -> B e. |^|(A \ suc B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   e. wcel 1300  A.wral 2105   \ cdif 2590   C_ wss 2593  |^|cint 3214  Oncon0 3657  suc csuc 3659

This theorem is referenced by:  unblem3 5635

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663

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