Theorem unexd 7741

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 7730	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
4	1, 2, 3	syl2anc 595	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2145  Vcvv 3457   ∪ cun 3905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3912  df-ss 3924  df-sn 4586  df-pr 4588  df-uni 4868

This theorem is referenced by:  sexp2  8130  sexp3  8137  mapunen  9122  sltsun1  27935  sltsun2  27936  addsproplem2  28117  addsuniflem  28148  sltmuls1  28294  sltmuls2  28295  precsexlem11  28364  suppun2  32937  elrgspnsubrunlem1  33475  elrgspnsubrunlem2  33476  elrgspnsubrun  33477  elrspunsn  33648  ofun  42861  tfsconcatun  43921  rclexi  44198  rtrclexlem  44199  trclubgNEW  44201  cnvrcl0  44208  dfrtrcl5  44212  iunrelexp0  44285  relexpmulg  44293  relexp01min  44296  clnbgrval  48443