MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  djuex Structured version   Visualization version   GIF version

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 5300 . . . . . 6 {∅} ∈ V
32a1i 11 . . . . 5 (𝐵𝑊 → {∅} ∈ V)
4 xpexg 7457 . . . . 5 (({∅} ∈ V ∧ 𝐴𝑉) → ({∅} × 𝐴) ∈ V)
53, 4sylan 583 . . . 4 ((𝐵𝑊𝐴𝑉) → ({∅} × 𝐴) ∈ V)
65ancoms 462 . . 3 ((𝐴𝑉𝐵𝑊) → ({∅} × 𝐴) ∈ V)
7 snex 5300 . . . . 5 {1o} ∈ V
87a1i 11 . . . 4 (𝐴𝑉 → {1o} ∈ V)
9 xpexg 7457 . . . 4 (({1o} ∈ V ∧ 𝐵𝑊) → ({1o} × 𝐵) ∈ V)
108, 9sylan 583 . . 3 ((𝐴𝑉𝐵𝑊) → ({1o} × 𝐵) ∈ V)
11 unexg 7456 . . 3 ((({∅} × 𝐴) ∈ V ∧ ({1o} × 𝐵) ∈ V) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V)
126, 10, 11syl2anc 587 . 2 ((𝐴𝑉𝐵𝑊) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∈ V)
131, 12eqeltrid 2897 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2112  Vcvv 3444  cun 3882  c0 4246  {csn 4528   × cxp 5521  1oc1o 8082  cdju 9315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-opab 5096  df-xp 5529  df-rel 5530  df-dju 9318
This theorem is referenced by:  djuexb  9326  updjud  9351  dju1dif  9587  pwdjuen  9596  alephadd  9992  gchhar  10094
  Copyright terms: Public domain W3C validator