Theorem uniexg 4125

Description: The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent A ∈ 𝑉 instead of A ∈ V to make the theorem more general and thus shorten some proofs; obviously the universal class constant V is one possible substitution for class variable 𝑉. (Contributed by NM, 25-Nov-1994.)

Assertion

Ref	Expression
uniexg	⊢ (A ∈ 𝑉 → ∪ A ∈ V)

Proof of Theorem uniexg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 3563	. . 3 ⊢ (x = A → ∪ x = ∪ A)
2	1	eleq1d 2088	. 2 ⊢ (x = A → (∪ x ∈ V ↔ ∪ A ∈ V))
3		vex 2538	. . 3 ⊢ x ∈ V
4	3	uniex 4124	. 2 ⊢ ∪ x ∈ V
5	2, 4	vtoclg 2590	1 ⊢ (A ∈ 𝑉 → ∪ A ∈ V)

Syntax hints:   → wi 4   = wceq 1228   ∈ wcel 1374  Vcvv 2535  ∪ cuni 3554

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-un 4120

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-uni 3555

This theorem is referenced by:  snnex  4131  uniexb  4155  ssonuni  4164  dmexg  4523  rnexg  4524  elxp4  4735  elxp5  4736  relrnfvex  5118  fvexg  5119  sefvex  5121  iunexg  5669  1stvalg  5692  2ndvalg  5693  cnvf1o  5769  brtpos2  5788  tfrlemiex  5866