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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 6928 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ⊔ 𝐵) ∈ V) | |
2 | df-dju 6923 | . . . . 5 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
3 | 2 | eleq1i 2205 | . . . 4 ⊢ ((𝐴 ⊔ 𝐵) ∈ V ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) |
4 | unexb 4363 | . . . 4 ⊢ ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V) | |
5 | 3, 4 | bitr4i 186 | . . 3 ⊢ ((𝐴 ⊔ 𝐵) ∈ V ↔ (({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V)) |
6 | 0ex 4055 | . . . . . . 7 ⊢ ∅ ∈ V | |
7 | 6 | snm 3643 | . . . . . 6 ⊢ ∃𝑥 𝑥 ∈ {∅} |
8 | rnxpm 4968 | . . . . . 6 ⊢ (∃𝑥 𝑥 ∈ {∅} → ran ({∅} × 𝐴) = 𝐴) | |
9 | 7, 8 | ax-mp 5 | . . . . 5 ⊢ ran ({∅} × 𝐴) = 𝐴 |
10 | rnexg 4804 | . . . . 5 ⊢ (({∅} × 𝐴) ∈ V → ran ({∅} × 𝐴) ∈ V) | |
11 | 9, 10 | eqeltrrid 2227 | . . . 4 ⊢ (({∅} × 𝐴) ∈ V → 𝐴 ∈ V) |
12 | 1oex 6321 | . . . . . . 7 ⊢ 1o ∈ V | |
13 | 12 | snm 3643 | . . . . . 6 ⊢ ∃𝑥 𝑥 ∈ {1o} |
14 | rnxpm 4968 | . . . . . 6 ⊢ (∃𝑥 𝑥 ∈ {1o} → ran ({1o} × 𝐵) = 𝐵) | |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ ran ({1o} × 𝐵) = 𝐵 |
16 | rnexg 4804 | . . . . 5 ⊢ (({1o} × 𝐵) ∈ V → ran ({1o} × 𝐵) ∈ V) | |
17 | 15, 16 | eqeltrrid 2227 | . . . 4 ⊢ (({1o} × 𝐵) ∈ V → 𝐵 ∈ V) |
18 | 11, 17 | anim12i 336 | . . 3 ⊢ ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
19 | 5, 18 | sylbi 120 | . 2 ⊢ ((𝐴 ⊔ 𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
20 | 1, 19 | impbii 125 | 1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ⊔ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1331 ∃wex 1468 ∈ wcel 1480 Vcvv 2686 ∪ cun 3069 ∅c0 3363 {csn 3527 × cxp 4537 ran crn 4540 1oc1o 6306 ⊔ cdju 6922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-tr 4027 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-dm 4549 df-rn 4550 df-1o 6313 df-dju 6923 |
This theorem is referenced by: ctfoex 7003 |
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