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Theorem djuex 9946
Description: The disjoint union of sets is a set. For a shorter proof using djuss 9958 see djuexALT 9960. (Contributed by AV, 28-Jun-2022.)
Assertion
Ref Expression
djuex ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Proof of Theorem djuex
StepHypRef Expression
1 df-dju 9939 . 2 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
2 snex 5442 . . . . . 6 {∅} ∈ V
32a1i 11 . . . . 5 (𝐵𝑊 → {∅} ∈ V)
4 xpexg 7769 . . . . 5 (({∅} ∈ V ∧ 𝐴𝑉) → ({∅} × 𝐴) ∈ V)
53, 4sylan 580 . . . 4 ((𝐵𝑊𝐴𝑉) → ({∅} × 𝐴) ∈ V)
65ancoms 458 . . 3 ((𝐴𝑉𝐵𝑊) → ({∅} × 𝐴) ∈ V)
7 snex 5442 . . . . 5 {1o} ∈ V
87a1i 11 . . . 4 (𝐴𝑉 → {1o} ∈ V)
9 xpexg 7769 . . . 4 (({1o} ∈ V ∧ 𝐵𝑊) → ({1o} × 𝐵) ∈ V)
108, 9sylan 580 . . 3 ((𝐴𝑉𝐵𝑊) → ({1o} × 𝐵) ∈ V)
11 unexg 7762 . . 3 ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V)
126, 10, 11syl2anc 584 . 2 ((𝐴𝑉𝐵𝑊) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V)
131, 12eqeltrid 2843 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  Vcvv 3478  cun 3961  c0 4339  {csn 4631   × cxp 5687  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-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-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-opab 5211  df-xp 5695  df-rel 5696  df-dju 9939
This theorem is referenced by:  djuexb  9947  updjud  9972  dju1dif  10211  pwdjuen  10220  alephadd  10615  gchhar  10717
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