Theorem onnmin 7003

Description: No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.)

Assertion

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

Proof of Theorem onnmin

Step	Hyp	Ref	Expression
1		intss1 4492	. . 3 |- ( B e. A -> |^| A C_ B )
2	1	adantl 482	. 2 |- ( ( A C_ On /\ B e. A ) -> |^| A C_ B )
3		ne0i 3921	. . . 4 |- ( B e. A -> A =/= (/) )
4		oninton 7000	. . . 4 |- ( ( A C_ On /\ A =/= (/) ) -> |^| A e. On )
5	3, 4	sylan2 491	. . 3 |- ( ( A C_ On /\ B e. A ) -> |^| A e. On )
6		ssel2 3598	. . 3 |- ( ( A C_ On /\ B e. A ) -> B e. On )
7		ontri1 5757	. . 3 |- ( ( |^| A e. On /\ B e. On ) -> ( |^| A C_ B <-> -. B e. |^| A ) )
8	5, 6, 7	syl2anc 693	. 2 |- ( ( A C_ On /\ B e. A ) -> ( |^| A C_ B <-> -. B e. |^| A ) )
9	2, 8	mpbid 222	1 |- ( ( A C_ On /\ B e. A ) -> -. B e. |^| A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   |^|cint 4475   Oncon0 5723

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-8 1992  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  ax-un 6949

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-tp 4182  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

This theorem is referenced by:  onnminsb  7004  oneqmin  7005  onmindif2  7012  cardmin2  8824  ackbij1lem18  9059  cofsmo  9091  fin23lem26  9147