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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 3447 | . . . 4 ⊢ (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦}) | |
2 | 1 | unieqd 3591 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ {𝑥, 𝑦} = ∪ {𝐴, 𝑦}) |
3 | uneq1 3090 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦)) | |
4 | 2, 3 | eqeq12d 2054 | . 2 ⊢ (𝑥 = 𝐴 → (∪ {𝑥, 𝑦} = (𝑥 ∪ 𝑦) ↔ ∪ {𝐴, 𝑦} = (𝐴 ∪ 𝑦))) |
5 | preq2 3448 | . . . 4 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵}) | |
6 | 5 | unieqd 3591 | . . 3 ⊢ (𝑦 = 𝐵 → ∪ {𝐴, 𝑦} = ∪ {𝐴, 𝐵}) |
7 | uneq2 3091 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵)) | |
8 | 6, 7 | eqeq12d 2054 | . 2 ⊢ (𝑦 = 𝐵 → (∪ {𝐴, 𝑦} = (𝐴 ∪ 𝑦) ↔ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))) |
9 | vex 2560 | . . 3 ⊢ 𝑥 ∈ V | |
10 | vex 2560 | . . 3 ⊢ 𝑦 ∈ V | |
11 | 9, 10 | unipr 3594 | . 2 ⊢ ∪ {𝑥, 𝑦} = (𝑥 ∪ 𝑦) |
12 | 4, 8, 11 | vtocl2g 2617 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 ∪ cun 2915 {cpr 3376 ∪ cuni 3580 |
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-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-uni 3581 |
This theorem is referenced by: onun2 4216 |
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