Theorem onmindif 3117

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 3038	. . . . . . . . . . 11 |- ((x e. On /\ B e. On) -> (x (_ B <-> -. B e. x))
2		onsssuc 3115	. . . . . . . . . . 11 |- ((x e. On /\ B e. On) -> (x (_ B <-> x e. suc B))
3	1, 2	bitr3d 541	. . . . . . . . . 10 |- ((x e. On /\ B e. On) -> (-. B e. x <-> x e. suc B))
4	3	con1bid 538	. . . . . . . . 9 |- ((x e. On /\ B e. On) -> (-. x e. suc B <-> B e. x))
5		ssel2 2115	. . . . . . . . 9 |- ((A (_ On /\ x e. A) -> x e. On)
6	4, 5	sylan 459	. . . . . . . 8 |- (((A (_ On /\ x e. A) /\ B e. On) -> (-. x e. suc B <-> B e. x))
7	6	biimpd 160	. . . . . . 7 |- (((A (_ On /\ x e. A) /\ B e. On) -> (-. x e. suc B -> B e. x))
8	7	exp31 385	. . . . . 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 372	. . . 4 |- ((A (_ On /\ B e. On) -> ((x e. A /\ -. x e. suc B) -> B e. x))
11		eldif 2108	. . . 4 |- (x e. (A \ suc B) <-> (x e. A /\ -. x e. suc B))
12	10, 11	syl5ib 213	. . 3 |- ((A (_ On /\ B e. On) -> (x e. (A \ suc B) -> B e. x))
13	12	r19.21aiv 1760	. 2 |- ((A (_ On /\ B e. On) -> A.x e. (A \ suc B)B e. x)
14		elintg 2595	. . 3 |- (B e. On -> (B e. |^|(A \ suc B) <-> A.x e. (A \ suc B)B e. x))
15	14	adantl 397	. 2 |- ((A (_ On /\ B e. On) -> (B e. |^|(A \ suc B) <-> A.x e. (A \ suc B)B e. x))
16	13, 15	mpbird 203	1 |- ((A (_ On /\ B e. On) -> B e. |^|(A \ suc B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 153   /\ wa 230   e. wcel 999  A.wral 1692   \ cdif 2095   (_ wss 2098  |^|cint 2587  Oncon0 3005  suc csuc 3007

This theorem is referenced by:  unblem3 4605

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-pow 2798  ax-pr 2835

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3or 788  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-op 2468  df-uni 2558  df-int 2588  df-br 2675  df-opab 2722  df-tr 2736  df-eprel 2888  df-po 2896  df-so 2906  df-fr 2974  df-we 2991  df-ord 3008  df-on 3009  df-suc 3011

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