Theorem onnmin 4659

Description: No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.) (Constructive proof by Mario Carneiro and Jim Kingdon, 21-Jul-2019.)

Assertion

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

Proof of Theorem onnmin

Step	Hyp	Ref	Expression
1		intss1 3937	. . 3 |- ( B e. A -> |^| A C_ B )
2		elirr 4632	. . . 4 |- -. B e. B
3		ssel 3218	. . . 4 |- ( |^| A C_ B -> ( B e. |^| A -> B e. B ) )
4	2, 3	mtoi 668	. . 3 |- ( |^| A C_ B -> -. B e. |^| A )
5	1, 4	syl 14	. 2 |- ( B e. A -> -. B e. |^| A )
6	5	adantl 277	1 |- ( ( A C_ On /\ B e. A ) -> -. B e. |^| A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2200   C_ wss 3197   |^|cint 3922   Oncon0 4453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-setind 4628

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-sn 3672  df-int 3923

This theorem is referenced by: (None)