Theorem ordn2lp 4601

Description: An ordinal class cannot 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 4599	. 2 |- ( Ord A -> -. A e. A )
2		ordtr 4595	. . 3 |- ( Ord A -> Tr A )
3		trel 4309	. . 3 |- ( Tr A -> ( ( A e. B /\ B e. A ) -> A e. A ) )
4	2, 3	syl 16	. 2 |- ( Ord A -> ( ( A e. B /\ B e. A ) -> A e. A ) )
5	1, 4	mtod 170	1 |- ( Ord A -> -. ( A e. B /\ B e. A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   e. wcel 1725   Tr wtr 4302   Ord word 4580

This theorem is referenced by:  ordtri1  4614  ordnbtwn  4672  suc11  4685  smoord  6627  unblem1  7359  cantnfp1lem3  7636  cardprclem  7866

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-fr 4541  df-we 4543  df-ord 4584