Theorem nordeq 4521

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 4519	. . . 4 |- ( Ord A -> -. A e. A )
2		eleq1 2229	. . . . 5 |- ( A = B -> ( A e. A <-> B e. A ) )
3	2	notbid 657	. . . 4 |- ( A = B -> ( -. A e. A <-> -. B e. A ) )
4	1, 3	syl5ibcom 154	. . 3 |- ( Ord A -> ( A = B -> -. B e. A ) )
5	4	necon2ad 2393	. 2 |- ( Ord A -> ( B e. A -> A =/= B ) )
6	5	imp 123	1 |- ( ( Ord A /\ B e. A ) -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   =/= wne 2336   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-sn 3582

This theorem is referenced by:  phplem1  6818