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Theorem djuexb 9950
Description: The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.)
Assertion
Ref Expression
djuexb ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)

Proof of Theorem djuexb
StepHypRef Expression
1 djuex 9949 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
2 df-dju 9942 . . . . 5 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
32eleq1i 2831 . . . 4 ((𝐴𝐵) ∈ V ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V)
4 unexb 7768 . . . 4 ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V)
53, 4bitr4i 278 . . 3 ((𝐴𝐵) ∈ V ↔ (({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V))
6 0nep0 5357 . . . . . 6 ∅ ≠ {∅}
76necomi 2994 . . . . 5 {∅} ≠ ∅
8 rnexg 7925 . . . . . 6 (({∅} × 𝐴) ∈ V → ran ({∅} × 𝐴) ∈ V)
9 rnxp 6189 . . . . . . 7 ({∅} ≠ ∅ → ran ({∅} × 𝐴) = 𝐴)
109eleq1d 2825 . . . . . 6 ({∅} ≠ ∅ → (ran ({∅} × 𝐴) ∈ V ↔ 𝐴 ∈ V))
118, 10imbitrid 244 . . . . 5 ({∅} ≠ ∅ → (({∅} × 𝐴) ∈ V → 𝐴 ∈ V))
127, 11ax-mp 5 . . . 4 (({∅} × 𝐴) ∈ V → 𝐴 ∈ V)
13 1oex 8517 . . . . . 6 1o ∈ V
1413snnz 4775 . . . . 5 {1o} ≠ ∅
15 rnexg 7925 . . . . . 6 (({1o} × 𝐵) ∈ V → ran ({1o} × 𝐵) ∈ V)
16 rnxp 6189 . . . . . . 7 ({1o} ≠ ∅ → ran ({1o} × 𝐵) = 𝐵)
1716eleq1d 2825 . . . . . 6 ({1o} ≠ ∅ → (ran ({1o} × 𝐵) ∈ V ↔ 𝐵 ∈ V))
1815, 17imbitrid 244 . . . . 5 ({1o} ≠ ∅ → (({1o} × 𝐵) ∈ V → 𝐵 ∈ V))
1914, 18ax-mp 5 . . . 4 (({1o} × 𝐵) ∈ V → 𝐵 ∈ V)
2012, 19anim12i 613 . . 3 ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
215, 20sylbi 217 . 2 ((𝐴𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
221, 21impbii 209 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2107  wne 2939  Vcvv 3479  cun 3948  c0 4332  {csn 4625   × cxp 5682  ran crn 5685  1oc1o 8500  cdju 9939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-rn 5695  df-suc 6389  df-1o 8507  df-dju 9942
This theorem is referenced by:  djuinf  10230  pwdjudom  10256
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