Theorem ordeq 2955

Description: Equality theorem for the ordinal predicate.

Assertion

Ref	Expression
ordeq	|- (A = B -> (Ord A <-> Ord B))

Proof of Theorem ordeq

Step	Hyp	Ref	Expression
1		treq 2686	. . 3 |- (A = B -> (Tr A <-> Tr B))
2		weeq2 2938	. . 3 |- (A = B -> (E We A <-> E We B))
3	1, 2	anbi12d 628	. 2 |- (A = B -> ((Tr A /\ E We A) <-> (Tr B /\ E We B)))
4		df-ord 2951	. 2 |- (Ord A <-> (Tr A /\ E We A))
5		df-ord 2951	. 2 |- (Ord B <-> (Tr B /\ E We B))
6	3, 4, 5	3bitr4g 555	1 |- (A = B -> (Ord A <-> Ord B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956  Tr wtr 2680  Ecep 2830   We wwe 2916  Ord word 2947

This theorem is referenced by:  elong 2956  limeq 2960  ordelord 2970  ordeleqon 2990  ordsuc 3065  ordun 3081  ordzsl 3116  elom 3134  elomg 3135  tfrlem8 3918

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-12 968  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  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-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-tr 2681  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951

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