![]() |
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 9802 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ⊔ 𝐵) ∈ V) | |
2 | df-dju 9795 | . . . . 5 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
3 | 2 | eleq1i 2828 | . . . 4 ⊢ ((𝐴 ⊔ 𝐵) ∈ V ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) |
4 | unexb 7674 | . . . 4 ⊢ ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) | |
5 | 3, 4 | bitr4i 277 | . . 3 ⊢ ((𝐴 ⊔ 𝐵) ∈ V ↔ (({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V)) |
6 | 0nep0 5311 | . . . . . 6 ⊢ ∅ ≠ {∅} | |
7 | 6 | necomi 2996 | . . . . 5 ⊢ {∅} ≠ ∅ |
8 | rnexg 7833 | . . . . . 6 ⊢ (({∅} × 𝐴) ∈ V → ran ({∅} × 𝐴) ∈ V) | |
9 | rnxp 6120 | . . . . . . 7 ⊢ ({∅} ≠ ∅ → ran ({∅} × 𝐴) = 𝐴) | |
10 | 9 | eleq1d 2822 | . . . . . 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 8414 | . . . . . 6 ⊢ 1o ∈ V | |
14 | 13 | snnz 4735 | . . . . 5 ⊢ {1o} ≠ ∅ |
15 | rnexg 7833 | . . . . . 6 ⊢ (({1o} × 𝐵) ∈ V → ran ({1o} × 𝐵) ∈ V) | |
16 | rnxp 6120 | . . . . . . 7 ⊢ ({1o} ≠ ∅ → ran ({1o} × 𝐵) = 𝐵) | |
17 | 16 | eleq1d 2822 | . . . . . 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 613 | . . 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 396 ∈ wcel 2106 ≠ wne 2941 Vcvv 3443 ∪ cun 3906 ∅c0 4280 {csn 4584 × cxp 5629 ran crn 5632 1oc1o 8397 ⊔ cdju 9792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-xp 5637 df-rel 5638 df-cnv 5639 df-dm 5641 df-rn 5642 df-suc 6321 df-1o 8404 df-dju 9795 |
This theorem is referenced by: djuinf 10082 pwdjudom 10110 |
Copyright terms: Public domain | W3C validator |