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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 9903 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ⊔ 𝐵) ∈ V) | |
2 | df-dju 9896 | . . . . 5 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
3 | 2 | eleq1i 2825 | . . . 4 ⊢ ((𝐴 ⊔ 𝐵) ∈ V ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) |
4 | unexb 7735 | . . . 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 5357 | . . . . . 6 ⊢ ∅ ≠ {∅} | |
7 | 6 | necomi 2996 | . . . . 5 ⊢ {∅} ≠ ∅ |
8 | rnexg 7895 | . . . . . 6 ⊢ (({∅} × 𝐴) ∈ V → ran ({∅} × 𝐴) ∈ V) | |
9 | rnxp 6170 | . . . . . . 7 ⊢ ({∅} ≠ ∅ → ran ({∅} × 𝐴) = 𝐴) | |
10 | 9 | eleq1d 2819 | . . . . . 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 8476 | . . . . . 6 ⊢ 1o ∈ V | |
14 | 13 | snnz 4781 | . . . . 5 ⊢ {1o} ≠ ∅ |
15 | rnexg 7895 | . . . . . 6 ⊢ (({1o} × 𝐵) ∈ V → ran ({1o} × 𝐵) ∈ V) | |
16 | rnxp 6170 | . . . . . . 7 ⊢ ({1o} ≠ ∅ → ran ({1o} × 𝐵) = 𝐵) | |
17 | 16 | eleq1d 2819 | . . . . . 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 2941 Vcvv 3475 ∪ cun 3947 ∅c0 4323 {csn 4629 × cxp 5675 ran crn 5678 1oc1o 8459 ⊔ cdju 9893 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-dm 5687 df-rn 5688 df-suc 6371 df-1o 8466 df-dju 9896 |
This theorem is referenced by: djuinf 10183 pwdjudom 10211 |
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