Theorem onnmin 4672

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 3948	. . 3 |- ( B e. A -> |^| A C_ B )
2		elirr 4645	. . . 4 |- -. B e. B
3		ssel 3222	. . . 4 |- ( |^| A C_ B -> ( B e. |^| A -> B e. B ) )
4	2, 3	mtoi 670	. . 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 2202   C_ wss 3201   |^|cint 3933   Oncon0 4466

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-v 2805  df-dif 3203  df-in 3207  df-ss 3214  df-sn 3679  df-int 3934

This theorem is referenced by: (None)