Theorem onnmin 4615

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 3899	. . 3 |- ( B e. A -> |^| A C_ B )
2		elirr 4588	. . . 4 |- -. B e. B
3		ssel 3186	. . . 4 |- ( |^| A C_ B -> ( B e. |^| A -> B e. B ) )
4	2, 3	mtoi 665	. . 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 2175   C_ wss 3165   |^|cint 3884   Oncon0 4409

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186  ax-setind 4584

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-v 2773  df-dif 3167  df-in 3171  df-ss 3178  df-sn 3638  df-int 3885

This theorem is referenced by: (None)