Theorem nordeq 2957

Description: A member of an ordinal class is not equal to it.

Assertion

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

Proof of Theorem nordeq

Step	Hyp	Ref	Expression
1		eleq1 1526	. . . . 5 |- (A = B -> (A e. A <-> B e. A))
2	1	negbid 609	. . . 4 |- (A = B -> (-. A e. A <-> -. B e. A))
3		ordirr 2956	. . . 4 |- (Ord A -> -. A e. A)
4	2, 3	syl5cbi 209	. . 3 |- (Ord A -> (A = B -> -. B e. A))
5	4	necon2ad 1606	. 2 |- (Ord A -> (B e. A -> A =/= B))
6	5	imp 350	1 |- ((Ord A /\ B e. A) -> A =/= B)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955   =/= wne 1577  Ord word 2937

This theorem is referenced by:  phplem1 4488  php 4493

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-eprel 2821  df-fr 2907  df-we 2924  df-ord 2941

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