![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > iuneq2 | GIF version |
Description: Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.) |
Ref | Expression |
---|---|
iuneq2 | ⊢ (∀x ∈ A B = 𝐶 → ∪ x ∈ A B = ∪ x ∈ A 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ss2iun 3663 | . . 3 ⊢ (∀x ∈ A B ⊆ 𝐶 → ∪ x ∈ A B ⊆ ∪ x ∈ A 𝐶) | |
2 | ss2iun 3663 | . . 3 ⊢ (∀x ∈ A 𝐶 ⊆ B → ∪ x ∈ A 𝐶 ⊆ ∪ x ∈ A B) | |
3 | 1, 2 | anim12i 321 | . 2 ⊢ ((∀x ∈ A B ⊆ 𝐶 ∧ ∀x ∈ A 𝐶 ⊆ B) → (∪ x ∈ A B ⊆ ∪ x ∈ A 𝐶 ∧ ∪ x ∈ A 𝐶 ⊆ ∪ x ∈ A B)) |
4 | eqss 2954 | . . . 4 ⊢ (B = 𝐶 ↔ (B ⊆ 𝐶 ∧ 𝐶 ⊆ B)) | |
5 | 4 | ralbii 2324 | . . 3 ⊢ (∀x ∈ A B = 𝐶 ↔ ∀x ∈ A (B ⊆ 𝐶 ∧ 𝐶 ⊆ B)) |
6 | r19.26 2435 | . . 3 ⊢ (∀x ∈ A (B ⊆ 𝐶 ∧ 𝐶 ⊆ B) ↔ (∀x ∈ A B ⊆ 𝐶 ∧ ∀x ∈ A 𝐶 ⊆ B)) | |
7 | 5, 6 | bitri 173 | . 2 ⊢ (∀x ∈ A B = 𝐶 ↔ (∀x ∈ A B ⊆ 𝐶 ∧ ∀x ∈ A 𝐶 ⊆ B)) |
8 | eqss 2954 | . 2 ⊢ (∪ x ∈ A B = ∪ x ∈ A 𝐶 ↔ (∪ x ∈ A B ⊆ ∪ x ∈ A 𝐶 ∧ ∪ x ∈ A 𝐶 ⊆ ∪ x ∈ A B)) | |
9 | 3, 7, 8 | 3imtr4i 190 | 1 ⊢ (∀x ∈ A B = 𝐶 → ∪ x ∈ A B = ∪ x ∈ A 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∀wral 2300 ⊆ wss 2911 ∪ ciun 3648 |
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-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-v 2553 df-in 2918 df-ss 2925 df-iun 3650 |
This theorem is referenced by: iuneq2i 3666 iuneq2dv 3669 dfmptg 5285 |
Copyright terms: Public domain | W3C validator |