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Mirrors > Home > ILE Home > Th. List > unexb | GIF version |
Description: Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.) |
Ref | Expression |
---|---|
unexb | ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq1 3223 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦)) | |
2 | 1 | eleq1d 2208 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝑦) ∈ V)) |
3 | uneq2 3224 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵)) | |
4 | 3 | eleq1d 2208 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝐵) ∈ V)) |
5 | vex 2689 | . . . 4 ⊢ 𝑥 ∈ V | |
6 | vex 2689 | . . . 4 ⊢ 𝑦 ∈ V | |
7 | 5, 6 | unex 4362 | . . 3 ⊢ (𝑥 ∪ 𝑦) ∈ V |
8 | 2, 4, 7 | vtocl2g 2750 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V) |
9 | ssun1 3239 | . . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
10 | ssexg 4067 | . . . 4 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐴 ∈ V) | |
11 | 9, 10 | mpan 420 | . . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐴 ∈ V) |
12 | ssun2 3240 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
13 | ssexg 4067 | . . . 4 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐵 ∈ V) | |
14 | 12, 13 | mpan 420 | . . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐵 ∈ V) |
15 | 11, 14 | jca 304 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
16 | 8, 15 | impbii 125 | 1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∪ cun 3069 ⊆ wss 3071 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-uni 3737 |
This theorem is referenced by: unexg 4364 sucexb 4413 frecabex 6295 djuexb 6929 |
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