Theorem unexd 7701

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 7690	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
4	1, 2, 3	syl2anc 585	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3430   ∪ cun 3888

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-un 3895  df-ss 3907  df-sn 4569  df-pr 4571  df-uni 4852

This theorem is referenced by:  sexp2  8089  sexp3  8096  sltsun1  27794  sltsun2  27795  addsproplem2  27976  addsuniflem  28007  sltmuls1  28153  sltmuls2  28154  precsexlem11  28223  suppun2  32772  elrgspnsubrunlem1  33323  elrgspnsubrunlem2  33324  elrgspnsubrun  33325  elrspunsn  33504  ofun  42691  tfsconcatun  43783  rclexi  44060  rtrclexlem  44061  trclubgNEW  44063  cnvrcl0  44070  dfrtrcl5  44074  iunrelexp0  44147  relexpmulg  44155  relexp01min  44158  clnbgrval  48310