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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 7069. (Contributed by AV, 28-Jun-2022.) |
Ref | Expression |
---|---|
djuex | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dju 7037 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | p0ex 4189 | . . . . . . 7 ⊢ {∅} ∈ V | |
3 | 2 | a1i 9 | . . . . . 6 ⊢ (𝐵 ∈ 𝑊 → {∅} ∈ V) |
4 | 3 | anim1i 340 | . . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ({∅} ∈ V ∧ 𝐴 ∈ 𝑉)) |
5 | 4 | ancoms 268 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({∅} ∈ V ∧ 𝐴 ∈ 𝑉)) |
6 | xpexg 4741 | . . . 4 ⊢ (({∅} ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} × 𝐴) ∈ V) | |
7 | 5, 6 | syl 14 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({∅} × 𝐴) ∈ V) |
8 | 1on 6424 | . . . . . . 7 ⊢ 1o ∈ On | |
9 | 8 | elexi 2750 | . . . . . 6 ⊢ 1o ∈ V |
10 | 9 | snex 4186 | . . . . 5 ⊢ {1o} ∈ V |
11 | 10 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {1o} ∈ V) |
12 | xpexg 4741 | . . . 4 ⊢ (({1o} ∈ V ∧ 𝐵 ∈ 𝑊) → ({1o} × 𝐵) ∈ V) | |
13 | 11, 12 | sylan 283 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({1o} × 𝐵) ∈ V) |
14 | unexg 4444 | . . 3 ⊢ ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) | |
15 | 7, 13, 14 | syl2anc 411 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) |
16 | 1, 15 | eqeltrid 2264 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 Vcvv 2738 ∪ cun 3128 ∅c0 3423 {csn 3593 Oncon0 4364 × cxp 4625 1oc1o 6410 ⊔ cdju 7036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2740 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-opab 4066 df-tr 4103 df-iord 4367 df-on 4369 df-suc 4372 df-xp 4633 df-1o 6417 df-dju 7037 |
This theorem is referenced by: djuexb 7043 updjud 7081 djudom 7092 exmidfodomrlemr 7201 exmidfodomrlemrALT 7202 djudoml 7218 djudomr 7219 exmidsbthrlem 14773 sbthom 14777 |
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