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Theorem limeq 4114
 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

Proof of Theorem limeq
StepHypRef Expression
1 ordeq 4109 . . 3
2 eleq2 2101 . . 3
3 id 19 . . . 4
4 unieq 3589 . . . 4
53, 4eqeq12d 2054 . . 3
61, 2, 53anbi123d 1207 . 2
7 dflim2 4107 . 2
8 dflim2 4107 . 2
96, 7, 83bitr4g 212 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 98   w3a 885   wceq 1243   wcel 1393  c0 3224  cuni 3580   word 4099   wlim 4101 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103  df-ilim 4106 This theorem is referenced by:  limuni2  4134
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