Theorem ordn2lp 4636

Description: An ordinal class cannot be an element of one of its members. Variant of first part of Theorem 2.2(vii) of [BellMachover] p. 469. (Contributed by NM, 3-Apr-1994.)

Assertion

Ref	Expression
ordn2lp	|- ( Ord A -> -. ( A e. B /\ B e. A ) )

Proof of Theorem ordn2lp

Step	Hyp	Ref	Expression
1		ordirr 4633	. 2 |- ( Ord A -> -. A e. A )
2		ordtr 4468	. . 3 |- ( Ord A -> Tr A )
3		trel 4188	. . 3 |- ( Tr A -> ( ( A e. B /\ B e. A ) -> A e. A ) )
4	2, 3	syl 14	. 2 |- ( Ord A -> ( ( A e. B /\ B e. A ) -> A e. A ) )
5	1, 4	mtod 667	1 |- ( Ord A -> -. ( A e. B /\ B e. A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2200   Tr wtr 4181   Ord word 4452

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-uni 3888  df-tr 4182  df-iord 4456

This theorem is referenced by:  nnnninfeq  7291