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Theorem djuexb 9372
 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 9371 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
2 df-dju 9364 . . . . 5 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
32eleq1i 2843 . . . 4 ((𝐴𝐵) ∈ V ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V)
4 unexb 7470 . . . 4 ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) ↔ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V)
53, 4bitr4i 281 . . 3 ((𝐴𝐵) ∈ V ↔ (({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V))
6 0nep0 5227 . . . . . 6 ∅ ≠ {∅}
76necomi 3006 . . . . 5 {∅} ≠ ∅
8 rnexg 7615 . . . . . 6 (({∅} × 𝐴) ∈ V → ran ({∅} × 𝐴) ∈ V)
9 rnxp 6000 . . . . . . 7 ({∅} ≠ ∅ → ran ({∅} × 𝐴) = 𝐴)
109eleq1d 2837 . . . . . 6 ({∅} ≠ ∅ → (ran ({∅} × 𝐴) ∈ V ↔ 𝐴 ∈ V))
118, 10syl5ib 247 . . . . 5 ({∅} ≠ ∅ → (({∅} × 𝐴) ∈ V → 𝐴 ∈ V))
127, 11ax-mp 5 . . . 4 (({∅} × 𝐴) ∈ V → 𝐴 ∈ V)
13 1oex 8121 . . . . . 6 1o ∈ V
1413snnz 4670 . . . . 5 {1o} ≠ ∅
15 rnexg 7615 . . . . . 6 (({1o} × 𝐵) ∈ V → ran ({1o} × 𝐵) ∈ V)
16 rnxp 6000 . . . . . . 7 ({1o} ≠ ∅ → ran ({1o} × 𝐵) = 𝐵)
1716eleq1d 2837 . . . . . 6 ({1o} ≠ ∅ → (ran ({1o} × 𝐵) ∈ V ↔ 𝐵 ∈ V))
1815, 17syl5ib 247 . . . . 5 ({1o} ≠ ∅ → (({1o} × 𝐵) ∈ V → 𝐵 ∈ V))
1914, 18ax-mp 5 . . . 4 (({1o} × 𝐵) ∈ V → 𝐵 ∈ V)
2012, 19anim12i 616 . . 3 ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
215, 20sylbi 220 . 2 ((𝐴𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
221, 21impbii 212 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2112   ≠ wne 2952  Vcvv 3410   ∪ cun 3857  ∅c0 4226  {csn 4523   × cxp 5523  ran crn 5526  1oc1o 8106   ⊔ cdju 9361 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-tr 5140  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-dm 5535  df-rn 5536  df-ord 6173  df-on 6174  df-suc 6176  df-1o 8113  df-dju 9364 This theorem is referenced by:  djuinf  9649  pwdjudom  9677
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