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Mirrors > Home > MPE Home > Th. List > djuexb | Structured version Visualization version 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 9946 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ⊔ 𝐵) ∈ V) | |
2 | df-dju 9939 | . . . . 5 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
3 | 2 | eleq1i 2830 | . . . 4 ⊢ ((𝐴 ⊔ 𝐵) ∈ V ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) |
4 | unexb 7766 | . . . 4 ⊢ ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) | |
5 | 3, 4 | bitr4i 278 | . . 3 ⊢ ((𝐴 ⊔ 𝐵) ∈ V ↔ (({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V)) |
6 | 0nep0 5364 | . . . . . 6 ⊢ ∅ ≠ {∅} | |
7 | 6 | necomi 2993 | . . . . 5 ⊢ {∅} ≠ ∅ |
8 | rnexg 7925 | . . . . . 6 ⊢ (({∅} × 𝐴) ∈ V → ran ({∅} × 𝐴) ∈ V) | |
9 | rnxp 6192 | . . . . . . 7 ⊢ ({∅} ≠ ∅ → ran ({∅} × 𝐴) = 𝐴) | |
10 | 9 | eleq1d 2824 | . . . . . 6 ⊢ ({∅} ≠ ∅ → (ran ({∅} × 𝐴) ∈ V ↔ 𝐴 ∈ V)) |
11 | 8, 10 | imbitrid 244 | . . . . 5 ⊢ ({∅} ≠ ∅ → (({∅} × 𝐴) ∈ V → 𝐴 ∈ V)) |
12 | 7, 11 | ax-mp 5 | . . . 4 ⊢ (({∅} × 𝐴) ∈ V → 𝐴 ∈ V) |
13 | 1oex 8515 | . . . . . 6 ⊢ 1o ∈ V | |
14 | 13 | snnz 4781 | . . . . 5 ⊢ {1o} ≠ ∅ |
15 | rnexg 7925 | . . . . . 6 ⊢ (({1o} × 𝐵) ∈ V → ran ({1o} × 𝐵) ∈ V) | |
16 | rnxp 6192 | . . . . . . 7 ⊢ ({1o} ≠ ∅ → ran ({1o} × 𝐵) = 𝐵) | |
17 | 16 | eleq1d 2824 | . . . . . 6 ⊢ ({1o} ≠ ∅ → (ran ({1o} × 𝐵) ∈ V ↔ 𝐵 ∈ V)) |
18 | 15, 17 | imbitrid 244 | . . . . 5 ⊢ ({1o} ≠ ∅ → (({1o} × 𝐵) ∈ V → 𝐵 ∈ V)) |
19 | 14, 18 | ax-mp 5 | . . . 4 ⊢ (({1o} × 𝐵) ∈ V → 𝐵 ∈ V) |
20 | 12, 19 | anim12i 613 | . . 3 ⊢ ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
21 | 5, 20 | sylbi 217 | . 2 ⊢ ((𝐴 ⊔ 𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
22 | 1, 21 | impbii 209 | 1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ⊔ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∪ cun 3961 ∅c0 4339 {csn 4631 × cxp 5687 ran crn 5690 1oc1o 8498 ⊔ cdju 9936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-suc 6392 df-1o 8505 df-dju 9939 |
This theorem is referenced by: djuinf 10227 pwdjudom 10253 |
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