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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdunexb | GIF version |
Description: Bounded version of unexb 4414. (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 3264 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦)) | |
2 | 1 | eleq1d 2233 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝑦) ∈ V)) |
3 | uneq2 3265 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵)) | |
4 | 3 | eleq1d 2233 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝐵) ∈ V)) |
5 | vex 2724 | . . . 4 ⊢ 𝑥 ∈ V | |
6 | vex 2724 | . . . 4 ⊢ 𝑦 ∈ V | |
7 | 5, 6 | bj-unex 13642 | . . 3 ⊢ (𝑥 ∪ 𝑦) ∈ V |
8 | 2, 4, 7 | vtocl2g 2785 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V) |
9 | ssun1 3280 | . . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
10 | bdunex.bd1 | . . . . 5 ⊢ BOUNDED 𝐴 | |
11 | 10 | bdssexg 13627 | . . . 4 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐴 ∈ V) |
12 | 9, 11 | mpan 421 | . . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐴 ∈ V) |
13 | ssun2 3281 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
14 | bdunex.bd2 | . . . . 5 ⊢ BOUNDED 𝐵 | |
15 | 14 | bdssexg 13627 | . . . 4 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐵 ∈ V) |
16 | 13, 15 | mpan 421 | . . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐵 ∈ V) |
17 | 12, 16 | jca 304 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
18 | 8, 17 | impbii 125 | 1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 Vcvv 2721 ∪ cun 3109 ⊆ wss 3111 BOUNDED wbdc 13563 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-pr 4181 ax-un 4405 ax-bd0 13536 ax-bdor 13539 ax-bdex 13542 ax-bdeq 13543 ax-bdel 13544 ax-bdsb 13545 ax-bdsep 13607 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-uni 3784 df-bdc 13564 |
This theorem is referenced by: (None) |
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