Theorem nordeq 2967

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 1534	. . . . 5 |- (A = B -> (A e. A <-> B e. A))
2	1	negbid 611	. . . 4 |- (A = B -> (-. A e. A <-> -. B e. A))
3		ordirr 2966	. . . 4 |- (Ord A -> -. A e. A)
4	2, 3	syl5cbi 209	. . 3 |- (Ord A -> (A = B -> -. B e. A))
5	4	necon2ad 1614	. 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 956   e. wcel 958   =/= wne 1585  Ord word 2947

This theorem is referenced by:  phplem1 4508  php 4513

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-eprel 2832  df-fr 2917  df-we 2934  df-ord 2951

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