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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdunexb | GIF version |
Description: Bounded version of unexb 4223. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bdunex.bd1 | ⊢ BOUNDED 𝐴 |
bdunex.bd2 | ⊢ BOUNDED 𝐵 |
Ref | Expression |
---|---|
bdunexb | ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq1 3129 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦)) | |
2 | 1 | eleq1d 2151 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝑦) ∈ V)) |
3 | uneq2 3130 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵)) | |
4 | 3 | eleq1d 2151 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝐵) ∈ V)) |
5 | vex 2613 | . . . 4 ⊢ 𝑥 ∈ V | |
6 | vex 2613 | . . . 4 ⊢ 𝑦 ∈ V | |
7 | 5, 6 | bj-unex 10977 | . . 3 ⊢ (𝑥 ∪ 𝑦) ∈ V |
8 | 2, 4, 7 | vtocl2g 2671 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V) |
9 | ssun1 3145 | . . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
10 | bdunex.bd1 | . . . . 5 ⊢ BOUNDED 𝐴 | |
11 | 10 | bdssexg 10962 | . . . 4 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐴 ∈ V) |
12 | 9, 11 | mpan 415 | . . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐴 ∈ V) |
13 | ssun2 3146 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
14 | bdunex.bd2 | . . . . 5 ⊢ BOUNDED 𝐵 | |
15 | 14 | bdssexg 10962 | . . . 4 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐵 ∈ V) |
16 | 13, 15 | mpan 415 | . . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐵 ∈ V) |
17 | 12, 16 | jca 300 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
18 | 8, 17 | impbii 124 | 1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 = wceq 1285 ∈ wcel 1434 Vcvv 2610 ∪ cun 2980 ⊆ wss 2982 BOUNDED wbdc 10898 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-pr 3992 ax-un 4216 ax-bd0 10871 ax-bdor 10874 ax-bdex 10877 ax-bdeq 10878 ax-bdel 10879 ax-bdsb 10880 ax-bdsep 10942 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-rex 2359 df-v 2612 df-un 2986 df-in 2988 df-ss 2995 df-sn 3422 df-pr 3423 df-uni 3622 df-bdc 10899 |
This theorem is referenced by: (None) |
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