Theorem unexd 7753

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 7745	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
4	1, 2, 3	syl2anc 596	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3450   ∪ cun 3897

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 3904  df-ss 3916  df-sn 4585  df-pr 4587  df-uni 4868

This theorem is used by:  sexp2  8144  sexp3  8151  mapunen  9144  sltsun1  28053  sltsun2  28054  addsproplem2  28235  addsuniflem  28266  sltmuls1  28412  sltmuls2  28413  precsexlem11  28482  suppun2  33156  elrgspnsubrunlem1  33687  elrgspnsubrunlem2  33688  elrspunsn  33857  ofun  43105  tfsconcatun  44178  rclexi  44455  rtrclexlem  44456  trclubgNEW  44458  cnvrcl0  44465  dfrtrcl5  44469  iunrelexp0  44542  relexpmulg  44550  relexp01min  44553  clnbgrval  48738