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Mirrors > Home > MPE Home > Th. List > djuex | Structured version Visualization version GIF version |
Description: The disjoint union of sets is a set. For a shorter proof using djuss 9958 see djuexALT 9960. (Contributed by AV, 28-Jun-2022.) |
Ref | Expression |
---|---|
djuex | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dju 9939 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | snex 5442 | . . . . . 6 ⊢ {∅} ∈ V | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → {∅} ∈ V) |
4 | xpexg 7769 | . . . . 5 ⊢ (({∅} ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} × 𝐴) ∈ V) | |
5 | 3, 4 | sylan 580 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ({∅} × 𝐴) ∈ V) |
6 | 5 | ancoms 458 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({∅} × 𝐴) ∈ V) |
7 | snex 5442 | . . . . 5 ⊢ {1o} ∈ V | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {1o} ∈ V) |
9 | xpexg 7769 | . . . 4 ⊢ (({1o} ∈ V ∧ 𝐵 ∈ 𝑊) → ({1o} × 𝐵) ∈ V) | |
10 | 8, 9 | sylan 580 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({1o} × 𝐵) ∈ V) |
11 | unexg 7762 | . . 3 ⊢ ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) | |
12 | 6, 10, 11 | syl2anc 584 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) |
13 | 1, 12 | eqeltrid 2843 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 ∅c0 4339 {csn 4631 × cxp 5687 1oc1o 8498 ⊔ cdju 9936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-opab 5211 df-xp 5695 df-rel 5696 df-dju 9939 |
This theorem is referenced by: djuexb 9947 updjud 9972 dju1dif 10211 pwdjuen 10220 alephadd 10615 gchhar 10717 |
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