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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 4109 | . . 3 ⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵)) | |
2 | eleq2 2101 | . . 3 ⊢ (𝐴 = 𝐵 → (∅ ∈ 𝐴 ↔ ∅ ∈ 𝐵)) | |
3 | id 19 | . . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | |
4 | unieq 3589 | . . . 4 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵) | |
5 | 3, 4 | eqeq12d 2054 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = ∪ 𝐴 ↔ 𝐵 = ∪ 𝐵)) |
6 | 1, 2, 5 | 3anbi123d 1207 | . 2 ⊢ (𝐴 = 𝐵 → ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴) ↔ (Ord 𝐵 ∧ ∅ ∈ 𝐵 ∧ 𝐵 = ∪ 𝐵))) |
7 | dflim2 4107 | . 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴)) | |
8 | dflim2 4107 | . 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 885 = wceq 1243 ∈ wcel 1393 ∅c0 3224 ∪ cuni 3580 Ord word 4099 Lim 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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