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Theorem djuexb 9904
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 9903 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
2 df-dju 9896 . . . . 5 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
32eleq1i 2825 . . . 4 ((𝐴𝐵) ∈ V ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V)
4 unexb 7735 . . . 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 2996 . . . . 5 {∅} ≠ ∅
8 rnexg 7895 . . . . . 6 (({∅} × 𝐴) ∈ V → ran ({∅} × 𝐴) ∈ V)
9 rnxp 6170 . . . . . . 7 ({∅} ≠ ∅ → ran ({∅} × 𝐴) = 𝐴)
109eleq1d 2819 . . . . . 6 ({∅} ≠ ∅ → (ran ({∅} × 𝐴) ∈ V ↔ 𝐴 ∈ V))
118, 10imbitrid 243 . . . . 5 ({∅} ≠ ∅ → (({∅} × 𝐴) ∈ V → 𝐴 ∈ V))
127, 11ax-mp 5 . . . 4 (({∅} × 𝐴) ∈ V → 𝐴 ∈ V)
13 1oex 8476 . . . . . 6 1o ∈ V
1413snnz 4781 . . . . 5 {1o} ≠ ∅
15 rnexg 7895 . . . . . 6 (({1o} × 𝐵) ∈ V → ran ({1o} × 𝐵) ∈ V)
16 rnxp 6170 . . . . . . 7 ({1o} ≠ ∅ → ran ({1o} × 𝐵) = 𝐵)
1716eleq1d 2819 . . . . . 6 ({1o} ≠ ∅ → (ran ({1o} × 𝐵) ∈ V ↔ 𝐵 ∈ V))
1815, 17imbitrid 243 . . . . 5 ({1o} ≠ ∅ → (({1o} × 𝐵) ∈ V → 𝐵 ∈ V))
1914, 18ax-mp 5 . . . 4 (({1o} × 𝐵) ∈ V → 𝐵 ∈ V)
2012, 19anim12i 614 . . 3 ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
215, 20sylbi 216 . 2 ((𝐴𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
221, 21impbii 208 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ 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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