Theorem uniexg 4141

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 3580	. . 3 ⊢ (x = A → ∪ x = ∪ A)
2	1	eleq1d 2103	. 2 ⊢ (x = A → (∪ x ∈ V ↔ ∪ A ∈ V))
3		vex 2554	. . 3 ⊢ x ∈ V
4	3	uniex 4140	. 2 ⊢ ∪ x ∈ V
5	2, 4	vtoclg 2607	1 ⊢ (A ∈ 𝑉 → ∪ A ∈ V)

Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  Vcvv 2551  ∪ cuni 3571

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-uni 3572

This theorem is referenced by:  snnex  4147  uniexb  4171  ssonuni  4180  dmexg  4539  rnexg  4540  elxp4  4751  elxp5  4752  relrnfvex  5136  fvexg  5137  sefvex  5139  riotaexg  5415  iunexg  5688  1stvalg  5711  2ndvalg  5712  cnvf1o  5788  brtpos2  5807  tfrlemiex  5886  en1bg  6216  en1uniel  6220