Theorem ordeq 2961

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 2691	. . 3 |- (A = B -> (Tr A <-> Tr B))
2		weeq2 2944	. . 3 |- (A = B -> (E We A <-> E We B))
3	1, 2	anbi12d 630	. 2 |- (A = B -> ((Tr A /\ E We A) <-> (Tr B /\ E We B)))
4		df-ord 2957	. 2 |- (Ord A <-> (Tr A /\ E We A))
5		df-ord 2957	. 2 |- (Ord B <-> (Tr B /\ E We B))
6	3, 4, 5	3bitr4g 557	1 |- (A = B -> (Ord A <-> Ord B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958  Tr wtr 2685  Ecep 2836   We wwe 2922  Ord word 2953

This theorem is referenced by:  elong 2962  limeq 2966  ordelord 2976  ordeleqon 2996  ordsuc 3071  ordun 3087  ordzsl 3122  elom 3140  elomg 3141  tfrlem8 3924

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-tr 2686  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957

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