Theorem uniexd 4561

Description: Deduction version of the ZF Axiom of Union in class notation. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
uniexd.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
uniexd	⊢ (𝜑 → ∪ 𝐴 ∈ V)

Proof of Theorem uniexd

Step	Hyp	Ref	Expression
1		uniexd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		uniexg 4560	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
3	1, 2	syl 14	1 ⊢ (𝜑 → ∪ 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2203  Vcvv 2813  ∪ cuni 3914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2815  df-uni 3915

This theorem is referenced by:  supex2g  7324  ptex  13477  zrhval  14765  zrhvalg  14766  zrhex  14769