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Theorem djuex 9325
Description: The disjoint union of sets is a set. For a shorter proof using djuss 9337 see djuexALT 9339. (Contributed by AV, 28-Jun-2022.)
Assertion
Ref Expression
djuex ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Proof of Theorem djuex
StepHypRef Expression
1 df-dju 9318 . 2 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
2 snex 5322 . . . . . 6 {∅} ∈ V
32a1i 11 . . . . 5 (𝐵𝑊 → {∅} ∈ V)
4 xpexg 7462 . . . . 5 (({∅} ∈ V ∧ 𝐴𝑉) → ({∅} × 𝐴) ∈ V)
53, 4sylan 580 . . . 4 ((𝐵𝑊𝐴𝑉) → ({∅} × 𝐴) ∈ V)
65ancoms 459 . . 3 ((𝐴𝑉𝐵𝑊) → ({∅} × 𝐴) ∈ V)
7 snex 5322 . . . . 5 {1o} ∈ V
87a1i 11 . . . 4 (𝐴𝑉 → {1o} ∈ V)
9 xpexg 7462 . . . 4 (({1o} ∈ V ∧ 𝐵𝑊) → ({1o} × 𝐵) ∈ V)
108, 9sylan 580 . . 3 ((𝐴𝑉𝐵𝑊) → ({1o} × 𝐵) ∈ V)
11 unexg 7461 . . 3 ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V)
126, 10, 11syl2anc 584 . 2 ((𝐴𝑉𝐵𝑊) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V)
131, 12eqeltrid 2914 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  Vcvv 3492  cun 3931  c0 4288  {csn 4557   × cxp 5546  1oc1o 8084  cdju 9315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-opab 5120  df-xp 5554  df-rel 5555  df-dju 9318
This theorem is referenced by:  djuexb  9326  updjud  9351  dju1dif  9586  pwdjuen  9595  alephadd  9987  gchhar  10089
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