![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 9851 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ⊔ 𝐵) ∈ V) | |
2 | df-dju 9844 | . . . . 5 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
3 | 2 | eleq1i 2829 | . . . 4 ⊢ ((𝐴 ⊔ 𝐵) ∈ V ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) |
4 | unexb 7687 | . . . 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 5318 | . . . . . 6 ⊢ ∅ ≠ {∅} | |
7 | 6 | necomi 2999 | . . . . 5 ⊢ {∅} ≠ ∅ |
8 | rnexg 7846 | . . . . . 6 ⊢ (({∅} × 𝐴) ∈ V → ran ({∅} × 𝐴) ∈ V) | |
9 | rnxp 6127 | . . . . . . 7 ⊢ ({∅} ≠ ∅ → ran ({∅} × 𝐴) = 𝐴) | |
10 | 9 | eleq1d 2823 | . . . . . 6 ⊢ ({∅} ≠ ∅ → (ran ({∅} × 𝐴) ∈ V ↔ 𝐴 ∈ V)) |
11 | 8, 10 | imbitrid 243 | . . . . 5 ⊢ ({∅} ≠ ∅ → (({∅} × 𝐴) ∈ V → 𝐴 ∈ V)) |
12 | 7, 11 | ax-mp 5 | . . . 4 ⊢ (({∅} × 𝐴) ∈ V → 𝐴 ∈ V) |
13 | 1oex 8427 | . . . . . 6 ⊢ 1o ∈ V | |
14 | 13 | snnz 4742 | . . . . 5 ⊢ {1o} ≠ ∅ |
15 | rnexg 7846 | . . . . . 6 ⊢ (({1o} × 𝐵) ∈ V → ran ({1o} × 𝐵) ∈ V) | |
16 | rnxp 6127 | . . . . . . 7 ⊢ ({1o} ≠ ∅ → ran ({1o} × 𝐵) = 𝐵) | |
17 | 16 | eleq1d 2823 | . . . . . 6 ⊢ ({1o} ≠ ∅ → (ran ({1o} × 𝐵) ∈ V ↔ 𝐵 ∈ V)) |
18 | 15, 17 | imbitrid 243 | . . . . 5 ⊢ ({1o} ≠ ∅ → (({1o} × 𝐵) ∈ V → 𝐵 ∈ V)) |
19 | 14, 18 | ax-mp 5 | . . . 4 ⊢ (({1o} × 𝐵) ∈ V → 𝐵 ∈ V) |
20 | 12, 19 | anim12i 614 | . . 3 ⊢ ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
21 | 5, 20 | sylbi 216 | . 2 ⊢ ((𝐴 ⊔ 𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
22 | 1, 21 | impbii 208 | 1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ⊔ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ≠ wne 2944 Vcvv 3448 ∪ cun 3913 ∅c0 4287 {csn 4591 × cxp 5636 ran crn 5639 1oc1o 8410 ⊔ cdju 9841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-xp 5644 df-rel 5645 df-cnv 5646 df-dm 5648 df-rn 5649 df-suc 6328 df-1o 8417 df-dju 9844 |
This theorem is referenced by: djuinf 10131 pwdjudom 10159 |
Copyright terms: Public domain | W3C validator |