Theorem limeq 2211

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 2206	. . 3 |- (A = B -> (Ord A <-> Ord B))
2		cleq1 1107	. . . 4 |- (A = B -> (A = (/) <-> B = (/)))
3	2	negbid 463	. . 3 |- (A = B -> (-. A = (/) <-> -. B = (/)))
4		unieq 1927	. . . . 5 |- (A = B -> U.A = U.B)
5	4	cleq2d 1112	. . . 4 |- (A = B -> (A = U.A <-> A = U.B))
6		cleq1 1107	. . . 4 |- (A = B -> (A = U.B <-> B = U.B))
7	5, 6	bitrd 406	. . 3 |- (A = B -> (A = U.A <-> B = U.B))
8	1, 3, 7	bi3and 636	. 2 |- (A = B -> ((Ord A /\ -. A = (/) /\ A = U.A) <-> (Ord B /\ -. B = (/) /\ B = U.B)))
9		df-lim 2204	. 2 |- (Lim A <-> (Ord A /\ -. A = (/) /\ A = U.A))
10		df-lim 2204	. 2 |- (Lim B <-> (Ord B /\ -. B = (/) /\ B = U.B))
11	8, 9, 10	3bitr4g 428	1 |- (A = B -> (Lim A <-> Lim B))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ w3a 581   = wceq 1091  (/)c0 1707  U.cuni 1919  Ord word 2198  Lim wlim 2200

This theorem is referenced by:  0ellim 2285  limsuc 2361  dflim3 2368  dfom2 2374  limomss 2378  nnlim 2385  omssnlim 2386  limom 2387  tfinds2 2405  ssnlim 2407  tz7.44lem1 2965  tz7.44-2 2967  tz7.44-3 2968  rdglem2 2976  rdglimt 2986  limensuc 3402  elom3 3478  alephislim 3688

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-lim 2204

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