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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdunexb | GIF version |
Description: Bounded version of unexb 4177. (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 3090 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦)) | |
2 | 1 | eleq1d 2106 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝑦) ∈ V)) |
3 | uneq2 3091 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵)) | |
4 | 3 | eleq1d 2106 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝐵) ∈ V)) |
5 | vex 2560 | . . . 4 ⊢ 𝑥 ∈ V | |
6 | vex 2560 | . . . 4 ⊢ 𝑦 ∈ V | |
7 | 5, 6 | bj-unex 10039 | . . 3 ⊢ (𝑥 ∪ 𝑦) ∈ V |
8 | 2, 4, 7 | vtocl2g 2617 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V) |
9 | ssun1 3106 | . . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
10 | bdunex.bd1 | . . . . 5 ⊢ BOUNDED 𝐴 | |
11 | 10 | bdssexg 10024 | . . . 4 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐴 ∈ V) |
12 | 9, 11 | mpan 400 | . . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐴 ∈ V) |
13 | ssun2 3107 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
14 | bdunex.bd2 | . . . . 5 ⊢ BOUNDED 𝐵 | |
15 | 14 | bdssexg 10024 | . . . 4 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐵 ∈ V) |
16 | 13, 15 | mpan 400 | . . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐵 ∈ V) |
17 | 12, 16 | jca 290 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
18 | 8, 17 | impbii 117 | 1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 Vcvv 2557 ∪ cun 2915 ⊆ wss 2917 BOUNDED wbdc 9960 |
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-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-pr 3944 ax-un 4170 ax-bd0 9933 ax-bdor 9936 ax-bdex 9939 ax-bdeq 9940 ax-bdel 9941 ax-bdsb 9942 ax-bdsep 10004 |
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-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-uni 3581 df-bdc 9961 |
This theorem is referenced by: (None) |
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