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Mirrors > Home > ILE Home > Th. List > djuexb | 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 7039 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ⊔ 𝐵) ∈ V) | |
2 | df-dju 7034 | . . . . 5 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
3 | 2 | eleq1i 2243 | . . . 4 ⊢ ((𝐴 ⊔ 𝐵) ∈ V ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) |
4 | unexb 4441 | . . . 4 ⊢ ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) | |
5 | 3, 4 | bitr4i 187 | . . 3 ⊢ ((𝐴 ⊔ 𝐵) ∈ V ↔ (({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V)) |
6 | 0ex 4129 | . . . . . . 7 ⊢ ∅ ∈ V | |
7 | 6 | snm 3712 | . . . . . 6 ⊢ ∃𝑥 𝑥 ∈ {∅} |
8 | rnxpm 5057 | . . . . . 6 ⊢ (∃𝑥 𝑥 ∈ {∅} → ran ({∅} × 𝐴) = 𝐴) | |
9 | 7, 8 | ax-mp 5 | . . . . 5 ⊢ ran ({∅} × 𝐴) = 𝐴 |
10 | rnexg 4891 | . . . . 5 ⊢ (({∅} × 𝐴) ∈ V → ran ({∅} × 𝐴) ∈ V) | |
11 | 9, 10 | eqeltrrid 2265 | . . . 4 ⊢ (({∅} × 𝐴) ∈ V → 𝐴 ∈ V) |
12 | 1oex 6422 | . . . . . . 7 ⊢ 1o ∈ V | |
13 | 12 | snm 3712 | . . . . . 6 ⊢ ∃𝑥 𝑥 ∈ {1o} |
14 | rnxpm 5057 | . . . . . 6 ⊢ (∃𝑥 𝑥 ∈ {1o} → ran ({1o} × 𝐵) = 𝐵) | |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ ran ({1o} × 𝐵) = 𝐵 |
16 | rnexg 4891 | . . . . 5 ⊢ (({1o} × 𝐵) ∈ V → ran ({1o} × 𝐵) ∈ V) | |
17 | 15, 16 | eqeltrrid 2265 | . . . 4 ⊢ (({1o} × 𝐵) ∈ V → 𝐵 ∈ V) |
18 | 11, 17 | anim12i 338 | . . 3 ⊢ ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
19 | 5, 18 | sylbi 121 | . 2 ⊢ ((𝐴 ⊔ 𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
20 | 1, 19 | impbii 126 | 1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ⊔ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1353 ∃wex 1492 ∈ wcel 2148 Vcvv 2737 ∪ cun 3127 ∅c0 3422 {csn 3592 × cxp 4623 ran crn 4626 1oc1o 6407 ⊔ cdju 7033 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-tr 4101 df-iord 4365 df-on 4367 df-suc 4370 df-xp 4631 df-rel 4632 df-cnv 4633 df-dm 4635 df-rn 4636 df-1o 6414 df-dju 7034 |
This theorem is referenced by: ctfoex 7114 |
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