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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 9949 see djuexALT 9951. (Contributed by AV, 28-Jun-2022.) |
Ref | Expression |
---|---|
djuex | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dju 9930 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | snex 5435 | . . . . . 6 ⊢ {∅} ∈ V | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → {∅} ∈ V) |
4 | xpexg 7756 | . . . . 5 ⊢ (({∅} ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} × 𝐴) ∈ V) | |
5 | 3, 4 | sylan 578 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ({∅} × 𝐴) ∈ V) |
6 | 5 | ancoms 457 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({∅} × 𝐴) ∈ V) |
7 | snex 5435 | . . . . 5 ⊢ {1o} ∈ V | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {1o} ∈ V) |
9 | xpexg 7756 | . . . 4 ⊢ (({1o} ∈ V ∧ 𝐵 ∈ 𝑊) → ({1o} × 𝐵) ∈ V) | |
10 | 8, 9 | sylan 578 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({1o} × 𝐵) ∈ V) |
11 | unexg 7755 | . . 3 ⊢ ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) | |
12 | 6, 10, 11 | syl2anc 582 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) |
13 | 1, 12 | eqeltrid 2832 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 Vcvv 3471 ∪ cun 3945 ∅c0 4324 {csn 4630 × cxp 5678 1oc1o 8484 ⊔ cdju 9927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-opab 5213 df-xp 5686 df-rel 5687 df-dju 9930 |
This theorem is referenced by: djuexb 9938 updjud 9963 dju1dif 10201 pwdjuen 10210 alephadd 10606 gchhar 10708 |
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