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Mirrors > Home > ILE Home > Th. List > djuex | GIF version |
Description: The disjoint union of sets is a set. See also the more precise djuss 6963. (Contributed by AV, 28-Jun-2022.) |
Ref | Expression |
---|---|
djuex | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dju 6931 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | p0ex 4120 | . . . . . . 7 ⊢ {∅} ∈ V | |
3 | 2 | a1i 9 | . . . . . 6 ⊢ (𝐵 ∈ 𝑊 → {∅} ∈ V) |
4 | 3 | anim1i 338 | . . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ({∅} ∈ V ∧ 𝐴 ∈ 𝑉)) |
5 | 4 | ancoms 266 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({∅} ∈ V ∧ 𝐴 ∈ 𝑉)) |
6 | xpexg 4661 | . . . 4 ⊢ (({∅} ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} × 𝐴) ∈ V) | |
7 | 5, 6 | syl 14 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({∅} × 𝐴) ∈ V) |
8 | 1on 6328 | . . . . . . 7 ⊢ 1o ∈ On | |
9 | 8 | elexi 2701 | . . . . . 6 ⊢ 1o ∈ V |
10 | 9 | snex 4117 | . . . . 5 ⊢ {1o} ∈ V |
11 | 10 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {1o} ∈ V) |
12 | xpexg 4661 | . . . 4 ⊢ (({1o} ∈ V ∧ 𝐵 ∈ 𝑊) → ({1o} × 𝐵) ∈ V) | |
13 | 11, 12 | sylan 281 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({1o} × 𝐵) ∈ V) |
14 | unexg 4372 | . . 3 ⊢ ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) | |
15 | 7, 13, 14 | syl2anc 409 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) |
16 | 1, 15 | eqeltrid 2227 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1481 Vcvv 2689 ∪ cun 3074 ∅c0 3368 {csn 3532 Oncon0 4293 × cxp 4545 1oc1o 6314 ⊔ cdju 6930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-opab 3998 df-tr 4035 df-iord 4296 df-on 4298 df-suc 4301 df-xp 4553 df-1o 6321 df-dju 6931 |
This theorem is referenced by: djuexb 6937 updjud 6975 djudom 6986 exmidfodomrlemr 7075 exmidfodomrlemrALT 7076 djudoml 7092 djudomr 7093 exmidsbthrlem 13392 sbthom 13396 |
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