Theorem nordeq 4528

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 4526	. . . 4 |- ( Ord A -> -. A e. A )
2		eleq1 2233	. . . . 5 |- ( A = B -> ( A e. A <-> B e. A ) )
3	2	notbid 662	. . . 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 2397	. 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 1348   e. wcel 2141   =/= wne 2340   Ord word 4347

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-v 2732  df-dif 3123  df-sn 3589

This theorem is referenced by:  phplem1  6830