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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 9321 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ⊔ 𝐵) ∈ V) | |
2 | df-dju 9314 | . . . . 5 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
3 | 2 | eleq1i 2880 | . . . 4 ⊢ ((𝐴 ⊔ 𝐵) ∈ V ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) |
4 | unexb 7451 | . . . 4 ⊢ ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) | |
5 | 3, 4 | bitr4i 281 | . . 3 ⊢ ((𝐴 ⊔ 𝐵) ∈ V ↔ (({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V)) |
6 | 0nep0 5223 | . . . . . 6 ⊢ ∅ ≠ {∅} | |
7 | 6 | necomi 3041 | . . . . 5 ⊢ {∅} ≠ ∅ |
8 | rnexg 7595 | . . . . . 6 ⊢ (({∅} × 𝐴) ∈ V → ran ({∅} × 𝐴) ∈ V) | |
9 | rnxp 5994 | . . . . . . 7 ⊢ ({∅} ≠ ∅ → ran ({∅} × 𝐴) = 𝐴) | |
10 | 9 | eleq1d 2874 | . . . . . 6 ⊢ ({∅} ≠ ∅ → (ran ({∅} × 𝐴) ∈ V ↔ 𝐴 ∈ V)) |
11 | 8, 10 | syl5ib 247 | . . . . 5 ⊢ ({∅} ≠ ∅ → (({∅} × 𝐴) ∈ V → 𝐴 ∈ V)) |
12 | 7, 11 | ax-mp 5 | . . . 4 ⊢ (({∅} × 𝐴) ∈ V → 𝐴 ∈ V) |
13 | 1oex 8093 | . . . . . 6 ⊢ 1o ∈ V | |
14 | 13 | snnz 4672 | . . . . 5 ⊢ {1o} ≠ ∅ |
15 | rnexg 7595 | . . . . . 6 ⊢ (({1o} × 𝐵) ∈ V → ran ({1o} × 𝐵) ∈ V) | |
16 | rnxp 5994 | . . . . . . 7 ⊢ ({1o} ≠ ∅ → ran ({1o} × 𝐵) = 𝐵) | |
17 | 16 | eleq1d 2874 | . . . . . 6 ⊢ ({1o} ≠ ∅ → (ran ({1o} × 𝐵) ∈ V ↔ 𝐵 ∈ V)) |
18 | 15, 17 | syl5ib 247 | . . . . 5 ⊢ ({1o} ≠ ∅ → (({1o} × 𝐵) ∈ V → 𝐵 ∈ V)) |
19 | 14, 18 | ax-mp 5 | . . . 4 ⊢ (({1o} × 𝐵) ∈ V → 𝐵 ∈ V) |
20 | 12, 19 | anim12i 615 | . . 3 ⊢ ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
21 | 5, 20 | sylbi 220 | . 2 ⊢ ((𝐴 ⊔ 𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
22 | 1, 21 | impbii 212 | 1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ⊔ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∪ cun 3879 ∅c0 4243 {csn 4525 × cxp 5517 ran crn 5520 1oc1o 8078 ⊔ cdju 9311 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-ord 6162 df-on 6163 df-suc 6165 df-1o 8085 df-dju 9314 |
This theorem is referenced by: djuinf 9599 pwdjudom 9627 |
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