Theorem limeq 2966

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 2961	. . 3 |- (A = B -> (Ord A <-> Ord B))
2		neeq1 1593	. . 3 |- (A = B -> (A =/= (/) <-> B =/= (/)))
3		unieq 2514	. . . . 5 |- (A = B -> U.A = U.B)
4	3	eqeq2d 1489	. . . 4 |- (A = B -> (A = U.A <-> A = U.B))
5		eqeq1 1484	. . . 4 |- (A = B -> (A = U.B <-> B = U.B))
6	4, 5	bitrd 530	. . 3 |- (A = B -> (A = U.A <-> B = U.B))
7	1, 2, 6	3anbi123d 895	. 2 |- (A = B -> ((Ord A /\ A =/= (/) /\ A = U.A) <-> (Ord B /\ B =/= (/) /\ B = U.B)))
8		df-lim 2959	. 2 |- (Lim A <-> (Ord A /\ A =/= (/) /\ A = U.A))
9		df-lim 2959	. 2 |- (Lim B <-> (Ord B /\ B =/= (/) /\ B = U.B))
10	7, 8, 9	3bitr4g 557	1 |- (A = B -> (Lim A <-> Lim B))

Syntax hints:   -> wi 3   <-> wb 146   /\ w3a 777   = wceq 958   =/= wne 1588  (/)c0 2283  U.cuni 2507  Ord word 2953  Lim wlim 2955

This theorem is referenced by:  limuni2 3036  0ellim 3037  dflim3 3124  limuni3 3129  dfom2 3139  limomss 3143  nnlim 3150  omssnlim 3151  limom 3152  tfinds2 3171  ssnlim 3173  tz7.44lem1 3933  tz7.44-2 3935  tz7.44-3 3936  dfrdg2 3939  rdglem2 3944  rdglimt 3954  limensuc 4513  elom3 4640  rankxplim2 4723  rankxplim3 4724  rankxpsuc 4725  alephislim 4894

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-tr 2686  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-lim 2959

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