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Theorem djuex 9948
Description: The disjoint union of sets is a set. For a shorter proof using djuss 9960 see djuexALT 9962. (Contributed by AV, 28-Jun-2022.)
Assertion
Ref Expression
djuex ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Proof of Theorem djuex
StepHypRef Expression
1 df-dju 9941 . 2 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
2 snex 5436 . . . . . 6 {∅} ∈ V
32a1i 11 . . . . 5 (𝐵𝑊 → {∅} ∈ V)
4 xpexg 7770 . . . . 5 (({∅} ∈ V ∧ 𝐴𝑉) → ({∅} × 𝐴) ∈ V)
53, 4sylan 580 . . . 4 ((𝐵𝑊𝐴𝑉) → ({∅} × 𝐴) ∈ V)
65ancoms 458 . . 3 ((𝐴𝑉𝐵𝑊) → ({∅} × 𝐴) ∈ V)
7 snex 5436 . . . . 5 {1o} ∈ V
87a1i 11 . . . 4 (𝐴𝑉 → {1o} ∈ V)
9 xpexg 7770 . . . 4 (({1o} ∈ V ∧ 𝐵𝑊) → ({1o} × 𝐵) ∈ V)
108, 9sylan 580 . . 3 ((𝐴𝑉𝐵𝑊) → ({1o} × 𝐵) ∈ V)
11 unexg 7763 . . 3 ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V)
126, 10, 11syl2anc 584 . 2 ((𝐴𝑉𝐵𝑊) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V)
131, 12eqeltrid 2845 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  Vcvv 3480  cun 3949  c0 4333  {csn 4626   × cxp 5683  1oc1o 8499  cdju 9938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-opab 5206  df-xp 5691  df-rel 5692  df-dju 9941
This theorem is referenced by:  djuexb  9949  updjud  9974  dju1dif  10213  pwdjuen  10222  alephadd  10617  gchhar  10719
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