Theorem unexd 7766

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 7758	. 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 3451   ∪ 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 2733  ax-sep 5249  ax-pr 5391  ax-un 7749

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 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-un 3904  df-ss 3916  df-sn 4585  df-pr 4587  df-uni 4868

This theorem is used by:  sexp2  8156  sexp3  8163  mapunen  9158  sltsun1  28167  sltsun2  28168  addsproplem2  28349  addsuniflem  28380  sltmuls1  28526  sltmuls2  28527  precsexlem11  28596  suppun2  33270  elrgspnsubrunlem1  33801  elrgspnsubrunlem2  33802  elrspunsn  33972  ofun  43269  tfsconcatun  44323  rclexi  44600  rtrclexlem  44601  trclubgNEW  44603  cnvrcl0  44610  dfrtrcl5  44614  iunrelexp0  44687  relexpmulg  44695  relexp01min  44698  clnbgrval  48889