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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 9081 see djuexALT 9083. (Contributed by AV, 28-Jun-2022.) |
Ref | Expression |
---|---|
djuex | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dju 9063 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | snex 5142 | . . . . . 6 ⊢ {∅} ∈ V | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → {∅} ∈ V) |
4 | xpexg 7239 | . . . . 5 ⊢ (({∅} ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} × 𝐴) ∈ V) | |
5 | 3, 4 | sylan 575 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ({∅} × 𝐴) ∈ V) |
6 | 5 | ancoms 452 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({∅} × 𝐴) ∈ V) |
7 | snex 5142 | . . . . 5 ⊢ {1o} ∈ V | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {1o} ∈ V) |
9 | xpexg 7239 | . . . 4 ⊢ (({1o} ∈ V ∧ 𝐵 ∈ 𝑊) → ({1o} × 𝐵) ∈ V) | |
10 | 8, 9 | sylan 575 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({1o} × 𝐵) ∈ V) |
11 | unexg 7238 | . . 3 ⊢ ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) | |
12 | 6, 10, 11 | syl2anc 579 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) |
13 | 1, 12 | syl5eqel 2863 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2107 Vcvv 3398 ∪ cun 3790 ∅c0 4141 {csn 4398 × cxp 5355 1oc1o 7838 ⊔ cdju 9060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-opab 4951 df-xp 5363 df-rel 5364 df-dju 9063 |
This theorem is referenced by: updjud 9095 |
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