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Theorem djuex 9937
Description: The disjoint union of sets is a set. For a shorter proof using djuss 9949 see djuexALT 9951. (Contributed by AV, 28-Jun-2022.)
Assertion
Ref Expression
djuex ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Proof of Theorem djuex
StepHypRef Expression
1 df-dju 9930 . 2 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
2 snex 5435 . . . . . 6 {∅} ∈ V
32a1i 11 . . . . 5 (𝐵𝑊 → {∅} ∈ V)
4 xpexg 7756 . . . . 5 (({∅} ∈ V ∧ 𝐴𝑉) → ({∅} × 𝐴) ∈ V)
53, 4sylan 578 . . . 4 ((𝐵𝑊𝐴𝑉) → ({∅} × 𝐴) ∈ V)
65ancoms 457 . . 3 ((𝐴𝑉𝐵𝑊) → ({∅} × 𝐴) ∈ V)
7 snex 5435 . . . . 5 {1o} ∈ V
87a1i 11 . . . 4 (𝐴𝑉 → {1o} ∈ V)
9 xpexg 7756 . . . 4 (({1o} ∈ V ∧ 𝐵𝑊) → ({1o} × 𝐵) ∈ V)
108, 9sylan 578 . . 3 ((𝐴𝑉𝐵𝑊) → ({1o} × 𝐵) ∈ V)
11 unexg 7755 . . 3 ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V)
126, 10, 11syl2anc 582 . 2 ((𝐴𝑉𝐵𝑊) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V)
131, 12eqeltrid 2832 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  Vcvv 3471  cun 3945  c0 4324  {csn 4630   × cxp 5678  1oc1o 8484  cdju 9927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-opab 5213  df-xp 5686  df-rel 5687  df-dju 9930
This theorem is referenced by:  djuexb  9938  updjud  9963  dju1dif  10201  pwdjuen  10210  alephadd  10606  gchhar  10708
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