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Theorem limeq 4373
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
StepHypRef Expression
1 ordeq 4368 . . 3 (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
2 eleq2 2241 . . 3 (𝐴 = 𝐵 → (∅ ∈ 𝐴 ↔ ∅ ∈ 𝐵))
3 id 19 . . . 4 (𝐴 = 𝐵𝐴 = 𝐵)
4 unieq 3816 . . . 4 (𝐴 = 𝐵 𝐴 = 𝐵)
53, 4eqeq12d 2192 . . 3 (𝐴 = 𝐵 → (𝐴 = 𝐴𝐵 = 𝐵))
61, 2, 53anbi123d 1312 . 2 (𝐴 = 𝐵 → ((Ord 𝐴 ∧ ∅ ∈ 𝐴𝐴 = 𝐴) ↔ (Ord 𝐵 ∧ ∅ ∈ 𝐵𝐵 = 𝐵)))
7 dflim2 4366 . 2 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴𝐴 = 𝐴))
8 dflim2 4366 . 2 (Lim 𝐵 ↔ (Ord 𝐵 ∧ ∅ ∈ 𝐵𝐵 = 𝐵))
96, 7, 83bitr4g 223 1 (𝐴 = 𝐵 → (Lim 𝐴 ↔ Lim 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  w3a 978   = wceq 1353  wcel 2148  c0 3422   cuni 3807  Ord word 4358  Lim wlim 4360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-in 3135  df-ss 3142  df-uni 3808  df-tr 4099  df-iord 4362  df-ilim 4365
This theorem is referenced by:  limuni2  4393
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