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Mirrors > Home > ILE Home > Th. List > djuexb | GIF version |
Description: The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.) |
Ref | Expression |
---|---|
djuexb | ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ⊔ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djuex 6999 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ⊔ 𝐵) ∈ V) | |
2 | df-dju 6994 | . . . . 5 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
3 | 2 | eleq1i 2230 | . . . 4 ⊢ ((𝐴 ⊔ 𝐵) ∈ V ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) |
4 | unexb 4414 | . . . 4 ⊢ ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) | |
5 | 3, 4 | bitr4i 186 | . . 3 ⊢ ((𝐴 ⊔ 𝐵) ∈ V ↔ (({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V)) |
6 | 0ex 4103 | . . . . . . 7 ⊢ ∅ ∈ V | |
7 | 6 | snm 3690 | . . . . . 6 ⊢ ∃𝑥 𝑥 ∈ {∅} |
8 | rnxpm 5027 | . . . . . 6 ⊢ (∃𝑥 𝑥 ∈ {∅} → ran ({∅} × 𝐴) = 𝐴) | |
9 | 7, 8 | ax-mp 5 | . . . . 5 ⊢ ran ({∅} × 𝐴) = 𝐴 |
10 | rnexg 4863 | . . . . 5 ⊢ (({∅} × 𝐴) ∈ V → ran ({∅} × 𝐴) ∈ V) | |
11 | 9, 10 | eqeltrrid 2252 | . . . 4 ⊢ (({∅} × 𝐴) ∈ V → 𝐴 ∈ V) |
12 | 1oex 6383 | . . . . . . 7 ⊢ 1o ∈ V | |
13 | 12 | snm 3690 | . . . . . 6 ⊢ ∃𝑥 𝑥 ∈ {1o} |
14 | rnxpm 5027 | . . . . . 6 ⊢ (∃𝑥 𝑥 ∈ {1o} → ran ({1o} × 𝐵) = 𝐵) | |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ ran ({1o} × 𝐵) = 𝐵 |
16 | rnexg 4863 | . . . . 5 ⊢ (({1o} × 𝐵) ∈ V → ran ({1o} × 𝐵) ∈ V) | |
17 | 15, 16 | eqeltrrid 2252 | . . . 4 ⊢ (({1o} × 𝐵) ∈ V → 𝐵 ∈ V) |
18 | 11, 17 | anim12i 336 | . . 3 ⊢ ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
19 | 5, 18 | sylbi 120 | . 2 ⊢ ((𝐴 ⊔ 𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
20 | 1, 19 | impbii 125 | 1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ⊔ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1342 ∃wex 1479 ∈ wcel 2135 Vcvv 2721 ∪ cun 3109 ∅c0 3404 {csn 3570 × cxp 4596 ran crn 4599 1oc1o 6368 ⊔ cdju 6993 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-tr 4075 df-iord 4338 df-on 4340 df-suc 4343 df-xp 4604 df-rel 4605 df-cnv 4606 df-dm 4608 df-rn 4609 df-1o 6375 df-dju 6994 |
This theorem is referenced by: ctfoex 7074 |
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