Theorem onmindif 5815

Description: When its successor is subtracted from a class of ordinal numbers, an ordinal number is less than the minimum of the resulting subclass. (Contributed by NM, 1-Dec-2003.)

Assertion

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

Proof of Theorem onmindif

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 3584	. . . 4 |- ( x e. ( A \ suc B ) <-> ( x e. A /\ -. x e. suc B ) )
2		ssel2 3598	. . . . . . . . 9 |- ( ( A C_ On /\ x e. A ) -> x e. On )
3		ontri1 5757	. . . . . . . . . . 11 |- ( ( x e. On /\ B e. On ) -> ( x C_ B <-> -. B e. x ) )
4		onsssuc 5813	. . . . . . . . . . 11 |- ( ( x e. On /\ B e. On ) -> ( x C_ B <-> x e. suc B ) )
5	3, 4	bitr3d 270	. . . . . . . . . 10 |- ( ( x e. On /\ B e. On ) -> ( -. B e. x <-> x e. suc B ) )
6	5	con1bid 345	. . . . . . . . 9 |- ( ( x e. On /\ B e. On ) -> ( -. x e. suc B <-> B e. x ) )
7	2, 6	sylan 488	. . . . . . . 8 |- ( ( ( A C_ On /\ x e. A ) /\ B e. On ) -> ( -. x e. suc B <-> B e. x ) )
8	7	biimpd 219	. . . . . . 7 |- ( ( ( A C_ On /\ x e. A ) /\ B e. On ) -> ( -. x e. suc B -> B e. x ) )
9	8	exp31 630	. . . . . 6 |- ( A C_ On -> ( x e. A -> ( B e. On -> ( -. x e. suc B -> B e. x ) ) ) )
10	9	com23 86	. . . . 5 |- ( A C_ On -> ( B e. On -> ( x e. A -> ( -. x e. suc B -> B e. x ) ) ) )
11	10	imp4b 613	. . . 4 |- ( ( A C_ On /\ B e. On ) -> ( ( x e. A /\ -. x e. suc B ) -> B e. x ) )
12	1, 11	syl5bi 232	. . 3 |- ( ( A C_ On /\ B e. On ) -> ( x e. ( A \ suc B ) -> B e. x ) )
13	12	ralrimiv 2965	. 2 |- ( ( A C_ On /\ B e. On ) -> A. x e. ( A \ suc B ) B e. x )
14		elintg 4483	. . 3 |- ( B e. On -> ( B e. |^| ( A \ suc B ) <-> A. x e. ( A \ suc B ) B e. x ) )
15	14	adantl 482	. 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 247	1 |- ( ( A C_ On /\ B e. On ) -> B e. |^| ( A \ suc B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912   \ cdif 3571   C_ wss 3574   |^|cint 4475   Oncon0 5723   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-suc 5729

This theorem is referenced by:  unblem3  8214  fin23lem26  9147