Theorem limeq 3669

Description: Equality theorem for the limit predicate. (The proof was shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

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

Proof of Theorem limeq

Step	Hyp	Ref	Expression
1		ordeq 3664	. . 3 |- (A = B -> (Ord A <-> Ord B))
2		neeq1 2024	. . 3 |- (A = B -> (A =/= (/) <-> B =/= (/)))
3		id 73	. . . 4 |- (A = B -> A = B)
4		unieq 3185	. . . 4 |- (A = B -> U.A = U.B)
5	3, 4	eqeq12d 1899	. . 3 |- (A = B -> (A = U.A <-> B = U.B))
6	1, 2, 5	3anbi123d 1168	. 2 |- (A = B -> ((Ord A /\ A =/= (/) /\ A = U.A) <-> (Ord B /\ B =/= (/) /\ B = U.B)))
7		df-lim 3662	. 2 |- (Lim A <-> (Ord A /\ A =/= (/) /\ A = U.A))
8		df-lim 3662	. 2 |- (Lim B <-> (Ord B /\ B =/= (/) /\ B = U.B))
9	6, 7, 8	3bitr4g 614	1 |- (A = B -> (Lim A <-> Lim B))

Syntax hints:   -> wi 3   <-> wb 163   /\ w3a 858   = wceq 1298   =/= wne 2017  (/)c0 2875  U.cuni 3177  Ord word 3656  Lim wlim 3658

This theorem is referenced by:  limuni2 3725  0ellim 3726  dflim3OLD 3931  limuni3 3936  tfinds2 3947  dfom2 3951  limomss 3956  nnlim 3964  omssnlimOLD 3966  limom 3967  ssnlim 3970  onfununi 5116  tz7.44lem1 5135  tz7.44-2 5137  tz7.44-3 5138  dfrdg2 5141  rdglem2 5146  rdglim 5156  limensuc 5601  elom3 5738  rankxplim2 5824  rankxplim3 5825  rankxpsuc 5826  omsublim 5887  alephislim 6031  cartarlim 15282  omsublimOLD 15396

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-in 2603  df-ss 2605  df-uni 3178  df-tr 3412  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-lim 3662

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