Theorem onmindif 3055

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 2976	. . . . . . . . . . 11 |- ((x e. On /\ B e. On) -> (x (_ B <-> -. B e. x))
2		onsssuc 3053	. . . . . . . . . . 11 |- ((x e. On /\ B e. On) -> (x (_ B <-> x e. suc B))
3	1, 2	bitr3d 529	. . . . . . . . . 10 |- ((x e. On /\ B e. On) -> (-. B e. x <-> x e. suc B))
4	3	con1bid 526	. . . . . . . . 9 |- ((x e. On /\ B e. On) -> (-. x e. suc B <-> B e. x))
5		ssel2 2060	. . . . . . . . 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 2053	. . . 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 1710	. 2 |- ((A (_ On /\ B e. On) -> A.x e. (A \ suc B)B e. x)
14		elintg 2536	. . 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 956  A.wral 1642   \ cdif 2040   (_ wss 2043  |^|cint 2528  Oncon0 2943  suc csuc 2945

This theorem is referenced by:  unblem3 4525

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-int 2529  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-suc 2949

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