Theorem nordeq 4636

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 4634	. . . 4 |- ( Ord A -> -. A e. A )
2		eleq1 2292	. . . . 5 |- ( A = B -> ( A e. A <-> B e. A ) )
3	2	notbid 671	. . . 4 |- ( A = B -> ( -. A e. A <-> -. B e. A ) )
4	1, 3	syl5ibcom 155	. . 3 |- ( Ord A -> ( A = B -> -. B e. A ) )
5	4	necon2ad 2457	. 2 |- ( Ord A -> ( B e. A -> A =/= B ) )
6	5	imp 124	1 |- ( ( Ord A /\ B e. A ) -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   =/= wne 2400   Ord word 4453

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-v 2801  df-dif 3199  df-sn 3672

This theorem is referenced by:  phplem1  7013