Theorem ordn2lp 4592

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 4589	. 2 |- ( Ord A -> -. A e. A )
2		ordtr 4424	. . 3 |- ( Ord A -> Tr A )
3		trel 4148	. . 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 664	1 |- ( Ord A -> -. ( A e. B /\ B e. A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2175   Tr wtr 4141   Ord word 4408

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-uni 3850  df-tr 4142  df-iord 4412

This theorem is referenced by:  nnnninfeq  7229