Theorem ordeq 2950

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 2681	. . 3 |- (A = B -> (Tr A <-> Tr B))
2		weeq2 2933	. . 3 |- (A = B -> (E We A <-> E We B))
3	1, 2	anbi12d 627	. 2 |- (A = B -> ((Tr A /\ E We A) <-> (Tr B /\ E We B)))
4		df-ord 2946	. 2 |- (Ord A <-> (Tr A /\ E We A))
5		df-ord 2946	. 2 |- (Ord B <-> (Tr B /\ E We B))
6	3, 4, 5	3bitr4g 554	1 |- (A = B -> (Ord A <-> Ord B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954  Tr wtr 2675  Ecep 2825   We wwe 2911  Ord word 2942

This theorem is referenced by:  elong 2951  limeq 2955  ordelord 2965  ordeleqon 2985  ordsuc 3060  ordun 3076  ordzsl 3111  elom 3129  elomg 3130  tfrlem8 3909

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-tr 2676  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946

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