Theorem ordn2lp 4611

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 4608	. 2 |- ( Ord A -> -. A e. A )
2		ordtr 4443	. . 3 |- ( Ord A -> Tr A )
3		trel 4165	. . 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 665	1 |- ( Ord A -> -. ( A e. B /\ B e. A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2178   Tr wtr 4158   Ord word 4427

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-v 2778  df-dif 3176  df-in 3180  df-ss 3187  df-sn 3649  df-uni 3865  df-tr 4159  df-iord 4431

This theorem is referenced by:  nnnninfeq  7256