Theorem uniexg 4356

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 3740	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
2	1	eleq1d 2206	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
3		vex 2684	. . 3 ⊢ 𝑥 ∈ V
4	3	uniex 4354	. 2 ⊢ ∪ 𝑥 ∈ V
5	2, 4	vtoclg 2741	1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480  Vcvv 2681  ∪ cuni 3731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-uni 3732

This theorem is referenced by:  abnexg  4362  snnex  4364  uniexb  4389  ssonuni  4399  dmexg  4798  rnexg  4799  elxp4  5021  elxp5  5022  relrnfvex  5432  fvexg  5433  sefvex  5435  riotaexg  5727  iunexg  6010  1stvalg  6033  2ndvalg  6034  cnvf1o  6115  brtpos2  6141  tfrlemiex  6221  tfr1onlemex  6237  tfrcllemex  6250  en1bg  6687  en1uniel  6691  fival  6851  suplocexprlem2b  7515  suplocexprlemlub  7525  restid  12120  istopon  12169  tgval  12207  tgvalex  12208  eltg  12210  eltg2  12211  tgss2  12237  ntrval  12268  restin  12334  cnovex  12354  cnprcl2k  12364  cnptopresti  12396  cnptoprest  12397  cnptoprest2  12398  lmtopcnp  12408  txbasex  12415  uptx  12432  reldvg  12806