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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 4289 | . . 3 ⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵)) | |
2 | eleq2 2201 | . . 3 ⊢ (𝐴 = 𝐵 → (∅ ∈ 𝐴 ↔ ∅ ∈ 𝐵)) | |
3 | id 19 | . . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | |
4 | unieq 3740 | . . . 4 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵) | |
5 | 3, 4 | eqeq12d 2152 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = ∪ 𝐴 ↔ 𝐵 = ∪ 𝐵)) |
6 | 1, 2, 5 | 3anbi123d 1290 | . 2 ⊢ (𝐴 = 𝐵 → ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴) ↔ (Ord 𝐵 ∧ ∅ ∈ 𝐵 ∧ 𝐵 = ∪ 𝐵))) |
7 | dflim2 4287 | . 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴)) | |
8 | dflim2 4287 | . 2 ⊢ (Lim 𝐵 ↔ (Ord 𝐵 ∧ ∅ ∈ 𝐵 ∧ 𝐵 = ∪ 𝐵)) | |
9 | 6, 7, 8 | 3bitr4g 222 | 1 ⊢ (𝐴 = 𝐵 → (Lim 𝐴 ↔ Lim 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 ∅c0 3358 ∪ cuni 3731 Ord word 4279 Lim wlim 4281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-in 3072 df-ss 3079 df-uni 3732 df-tr 4022 df-iord 4283 df-ilim 4286 |
This theorem is referenced by: limuni2 4314 |
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