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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdunexb | GIF version |
Description: Bounded version of unexb 4323. (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 3189 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦)) | |
2 | 1 | eleq1d 2183 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝑦) ∈ V)) |
3 | uneq2 3190 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵)) | |
4 | 3 | eleq1d 2183 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝐵) ∈ V)) |
5 | vex 2660 | . . . 4 ⊢ 𝑥 ∈ V | |
6 | vex 2660 | . . . 4 ⊢ 𝑦 ∈ V | |
7 | 5, 6 | bj-unex 12809 | . . 3 ⊢ (𝑥 ∪ 𝑦) ∈ V |
8 | 2, 4, 7 | vtocl2g 2721 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V) |
9 | ssun1 3205 | . . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
10 | bdunex.bd1 | . . . . 5 ⊢ BOUNDED 𝐴 | |
11 | 10 | bdssexg 12794 | . . . 4 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐴 ∈ V) |
12 | 9, 11 | mpan 418 | . . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐴 ∈ V) |
13 | ssun2 3206 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
14 | bdunex.bd2 | . . . . 5 ⊢ BOUNDED 𝐵 | |
15 | 14 | bdssexg 12794 | . . . 4 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐵 ∈ V) |
16 | 13, 15 | mpan 418 | . . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐵 ∈ V) |
17 | 12, 16 | jca 302 | . 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 1314 ∈ wcel 1463 Vcvv 2657 ∪ cun 3035 ⊆ wss 3037 BOUNDED wbdc 12730 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-pr 4091 ax-un 4315 ax-bd0 12703 ax-bdor 12706 ax-bdex 12709 ax-bdeq 12710 ax-bdel 12711 ax-bdsb 12712 ax-bdsep 12774 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-rex 2396 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-sn 3499 df-pr 3500 df-uni 3703 df-bdc 12731 |
This theorem is referenced by: (None) |
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