Theorem ordn2lp 4522

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 4519	. 2 |- ( Ord A -> -. A e. A )
2		ordtr 4356	. . 3 |- ( Ord A -> Tr A )
3		trel 4087	. . 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 653	1 |- ( Ord A -> -. ( A e. B /\ B e. A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   e. wcel 2136   Tr wtr 4080   Ord word 4340

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-sn 3582  df-uni 3790  df-tr 4081  df-iord 4344

This theorem is referenced by:  nnnninfeq  7092