Theorem onnmin 4552

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 3846	. . 3 |- ( B e. A -> |^| A C_ B )
2		elirr 4525	. . . 4 |- -. B e. B
3		ssel 3141	. . . 4 |- ( |^| A C_ B -> ( B e. |^| A -> B e. B ) )
4	2, 3	mtoi 659	. . 3 |- ( |^| A C_ B -> -. B e. |^| A )
5	1, 4	syl 14	. 2 |- ( B e. A -> -. B e. |^| A )
6	5	adantl 275	1 |- ( ( A C_ On /\ B e. A ) -> -. B e. |^| A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   e. wcel 2141   C_ wss 3121   |^|cint 3831   Oncon0 4348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-sn 3589  df-int 3832

This theorem is referenced by: (None)