Theorem uniexd 4531

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 4530	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
3	1, 2	syl 14	1 ⊢ (𝜑 → ∪ 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2200  Vcvv 2799  ∪ cuni 3888

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-uni 3889

This theorem is referenced by:  supex2g  7200  ptex  13297  zrhval  14581  zrhvalg  14582  zrhex  14585