Theorem nordeq 4610

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 4608	. . . 4 |- ( Ord A -> -. A e. A )
2		eleq1 2270	. . . . 5 |- ( A = B -> ( A e. A <-> B e. A ) )
3	2	notbid 669	. . . 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 2435	. 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 1373   e. wcel 2178   =/= wne 2378   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-sn 3649

This theorem is referenced by:  phplem1  6974