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Theorem djuexb 9947
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 9946 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
2 df-dju 9939 . . . . 5 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
32eleq1i 2830 . . . 4 ((𝐴𝐵) ∈ V ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V)
4 unexb 7766 . . . 4 ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V)
53, 4bitr4i 278 . . 3 ((𝐴𝐵) ∈ V ↔ (({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V))
6 0nep0 5364 . . . . . 6 ∅ ≠ {∅}
76necomi 2993 . . . . 5 {∅} ≠ ∅
8 rnexg 7925 . . . . . 6 (({∅} × 𝐴) ∈ V → ran ({∅} × 𝐴) ∈ V)
9 rnxp 6192 . . . . . . 7 ({∅} ≠ ∅ → ran ({∅} × 𝐴) = 𝐴)
109eleq1d 2824 . . . . . 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 8515 . . . . . 6 1o ∈ V
1413snnz 4781 . . . . 5 {1o} ≠ ∅
15 rnexg 7925 . . . . . 6 (({1o} × 𝐵) ∈ V → ran ({1o} × 𝐵) ∈ V)
16 rnxp 6192 . . . . . . 7 ({1o} ≠ ∅ → ran ({1o} × 𝐵) = 𝐵)
1716eleq1d 2824 . . . . . 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 2106  wne 2938  Vcvv 3478  cun 3961  c0 4339  {csn 4631   × cxp 5687  ran crn 5690  1oc1o 8498  cdju 9936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-suc 6392  df-1o 8505  df-dju 9939
This theorem is referenced by:  djuinf  10227  pwdjudom  10253
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