Theorem nordeq 4360

Description: A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)

Assertion

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

Proof of Theorem nordeq

Step	Hyp	Ref	Expression
1		ordirr 4358	. . . 4 |- ( Ord A -> -. A e. A )
2		eleq1 2150	. . . . 5 |- ( A = B -> ( A e. A <-> B e. A ) )
3	2	notbid 627	. . . 4 |- ( A = B -> ( -. A e. A <-> -. B e. A ) )
4	1, 3	syl5ibcom 153	. . 3 |- ( Ord A -> ( A = B -> -. B e. A ) )
5	4	necon2ad 2312	. 2 |- ( Ord A -> ( B e. A -> A =/= B ) )
6	5	imp 122	1 |- ( ( Ord A /\ B e. A ) -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   = wceq 1289   e. wcel 1438   =/= wne 2255   Ord word 4189

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-setind 4353

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 3001  df-sn 3452

This theorem is referenced by:  phplem1  6568