Theorem limeq 5435

Description: Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof 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 5430	. . 3 |- ( A = B -> ( Ord A <-> Ord B ) )
2		neeq1 2686	. . 3 |- ( A = B -> ( A =/= (/) <-> B =/= (/) ) )
3		id 22	. . . 4 |- ( A = B -> A = B )
4		unieq 4206	. . . 4 |- ( A = B -> U. A = U. B )
5	3, 4	eqeq12d 2466	. . 3 |- ( A = B -> ( A = U. A <-> B = U. B ) )
6	1, 2, 5	3anbi123d 1339	. 2 |- ( A = B -> ( ( Ord A /\ A =/= (/) /\ A = U. A ) <-> ( Ord B /\ B =/= (/) /\ B = U. B ) ) )
7		df-lim 5428	. 2 |- ( Lim A <-> ( Ord A /\ A =/= (/) /\ A = U. A ) )
8		df-lim 5428	. 2 |- ( Lim B <-> ( Ord B /\ B =/= (/) /\ B = U. B ) )
9	6, 7, 8	3bitr4g 292	1 |- ( A = B -> ( Lim A <-> Lim B ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ w3a 985   = wceq 1444   =/= wne 2622   (/)c0 3731   U.cuni 4198   Ord word 5422   Lim wlim 5424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-in 3411  df-ss 3418  df-uni 4199  df-tr 4498  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-ord 5426  df-lim 5428

This theorem is referenced by:  limuni2  5484  0ellim  5485  limuni3  6679  tfinds2  6690  dfom2  6694  limomss  6697  nnlim  6705  limom  6707  ssnlim  6710  onfununi  7060  tfr1a  7112  tz7.44lem1  7123  tz7.44-2  7125  tz7.44-3  7126  oeeulem  7302  limensuc  7749  elom3  8153  r1funlim  8237  rankxplim2  8351  rankxplim3  8352  rankxpsuc  8353  infxpenlem  8444  alephislim  8514  cflim2  8693  winalim  9120  rankcf  9202  gruina  9243  rdgprc0  30440  dfrdg2  30442  dfrdg4  30718  limsucncmpi  31105  limsucncmp  31106