Theorem unexd 7687

Description: The union of two sets is a set. (Contributed by SN, 16-Jul-2024.)

Hypotheses

Ref	Expression
unexd.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
unexd.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)

Assertion

Ref	Expression
unexd	⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)

Proof of Theorem unexd

Step	Hyp	Ref	Expression
1		unexd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		unexd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3		unexg 7676	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
4	1, 2, 3	syl2anc 584	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2111  Vcvv 3436   ∪ cun 3900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-sn 4577  df-pr 4579  df-uni 4860

This theorem is referenced by:  sexp2  8076  sexp3  8083  ssltun1  27750  ssltun2  27751  addsproplem2  27914  addsuniflem  27945  ssltmul1  28087  ssltmul2  28088  precsexlem11  28156  suppun2  32663  elrgspnsubrunlem1  33212  elrgspnsubrunlem2  33213  elrgspnsubrun  33214  elrspunsn  33392  ofun  42275  tfsconcatun  43376  rclexi  43654  rtrclexlem  43655  trclubgNEW  43657  cnvrcl0  43664  dfrtrcl5  43668  iunrelexp0  43741  relexpmulg  43749  relexp01min  43752  clnbgrval  47859