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Mirrors > Home > ILE Home > Th. List > limeq | GIF version |
Description: Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
limeq | ⊢ (A = B → (Lim A ↔ Lim B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordeq 4075 | . . 3 ⊢ (A = B → (Ord A ↔ Ord B)) | |
2 | eleq2 2098 | . . 3 ⊢ (A = B → (∅ ∈ A ↔ ∅ ∈ B)) | |
3 | id 19 | . . . 4 ⊢ (A = B → A = B) | |
4 | unieq 3580 | . . . 4 ⊢ (A = B → ∪ A = ∪ B) | |
5 | 3, 4 | eqeq12d 2051 | . . 3 ⊢ (A = B → (A = ∪ A ↔ B = ∪ B)) |
6 | 1, 2, 5 | 3anbi123d 1206 | . 2 ⊢ (A = B → ((Ord A ∧ ∅ ∈ A ∧ A = ∪ A) ↔ (Ord B ∧ ∅ ∈ B ∧ B = ∪ B))) |
7 | dflim2 4073 | . 2 ⊢ (Lim A ↔ (Ord A ∧ ∅ ∈ A ∧ A = ∪ A)) | |
8 | dflim2 4073 | . 2 ⊢ (Lim B ↔ (Ord B ∧ ∅ ∈ B ∧ B = ∪ B)) | |
9 | 6, 7, 8 | 3bitr4g 212 | 1 ⊢ (A = B → (Lim A ↔ Lim B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∧ w3a 884 = wceq 1242 ∈ wcel 1390 ∅c0 3218 ∪ cuni 3571 Ord word 4065 Lim wlim 4067 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-in 2918 df-ss 2925 df-uni 3572 df-tr 3846 df-iord 4069 df-ilim 4072 |
This theorem is referenced by: limuni2 4100 |
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