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Mirrors > Home > ILE Home > Th. List > uniprg | GIF version |
Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.) |
Ref | Expression |
---|---|
uniprg | ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ∪ {A, B} = (A ∪ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1 3438 | . . . 4 ⊢ (x = A → {x, y} = {A, y}) | |
2 | 1 | unieqd 3582 | . . 3 ⊢ (x = A → ∪ {x, y} = ∪ {A, y}) |
3 | uneq1 3084 | . . 3 ⊢ (x = A → (x ∪ y) = (A ∪ y)) | |
4 | 2, 3 | eqeq12d 2051 | . 2 ⊢ (x = A → (∪ {x, y} = (x ∪ y) ↔ ∪ {A, y} = (A ∪ y))) |
5 | preq2 3439 | . . . 4 ⊢ (y = B → {A, y} = {A, B}) | |
6 | 5 | unieqd 3582 | . . 3 ⊢ (y = B → ∪ {A, y} = ∪ {A, B}) |
7 | uneq2 3085 | . . 3 ⊢ (y = B → (A ∪ y) = (A ∪ B)) | |
8 | 6, 7 | eqeq12d 2051 | . 2 ⊢ (y = B → (∪ {A, y} = (A ∪ y) ↔ ∪ {A, B} = (A ∪ B))) |
9 | vex 2554 | . . 3 ⊢ x ∈ V | |
10 | vex 2554 | . . 3 ⊢ y ∈ V | |
11 | 9, 10 | unipr 3585 | . 2 ⊢ ∪ {x, y} = (x ∪ y) |
12 | 4, 8, 11 | vtocl2g 2611 | 1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ∪ {A, B} = (A ∪ B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 ∪ cun 2909 {cpr 3368 ∪ cuni 3571 |
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-rex 2306 df-v 2553 df-un 2916 df-sn 3373 df-pr 3374 df-uni 3572 |
This theorem is referenced by: onun2 4182 |
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