Theorem limeq 2987

Description: Equality theorem for the limit predicate.

Assertion

Ref	Expression
limeq	|- (A = B -> (Lim A <-> Lim B))

Proof of Theorem limeq

Step	Hyp	Ref	Expression
1		ordeq 2982	. . 3 |- (A = B -> (Ord A <-> Ord B))
2		neeq1 1633	. . 3 |- (A = B -> (A =/= (/) <-> B =/= (/)))
3		unieq 2576	. . . . 5 |- (A = B -> U.A = U.B)
4	3	eqeq2d 1529	. . . 4 |- (A = B -> (A = U.A <-> A = U.B))
5		eqeq1 1524	. . . 4 |- (A = B -> (A = U.B <-> B = U.B))
6	4, 5	bitrd 531	. . 3 |- (A = B -> (A = U.A <-> B = U.B))
7	1, 2, 6	3anbi123d 899	. 2 |- (A = B -> ((Ord A /\ A =/= (/) /\ A = U.A) <-> (Ord B /\ B =/= (/) /\ B = U.B)))
8		df-lim 2980	. 2 |- (Lim A <-> (Ord A /\ A =/= (/) /\ A = U.A))
9		df-lim 2980	. 2 |- (Lim B <-> (Ord B /\ B =/= (/) /\ B = U.B))
10	7, 8, 9	3bitr4g 558	1 |- (A = B -> (Lim A <-> Lim B))

Syntax hints:   -> wi 3   <-> wb 144   /\ w3a 781   = wceq 992   =/= wne 1628  (/)c0 2332  U.cuni 2569  Ord word 2974  Lim wlim 2976

This theorem is referenced by:  limuni2 3034  0ellim 3035  dflim3 3201  limuni3 3206  tfinds2 3216  dfom2 3220  limomss 3224  nnlim 3231  omssnlim 3232  limom 3233  ssnlim 3236  onfununi 4209  tz7.44lem1 4228  tz7.44-2 4230  tz7.44-3 4231  dfrdg2 4234  rdglem2 4239  rdglim 4249  limensuc 4654  elom3 4777  rankxplim2 4859  rankxplim3 4860  rankxpsuc 4861  alephislim 5033  omsublim 11448

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 783  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-sn 2470  df-pr 2471  df-op 2474  df-uni 2570  df-br 2693  df-tr 2755  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-lim 2980

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