Theorem uniexg 4534

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 3900	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
2	1	eleq1d 2298	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
3		vex 2803	. . 3 ⊢ 𝑥 ∈ V
4	3	uniex 4532	. 2 ⊢ ∪ 𝑥 ∈ V
5	2, 4	vtoclg 2862	1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  Vcvv 2800  ∪ cuni 3891

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 4205  ax-un 4528

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 2802  df-uni 3892

This theorem is referenced by:  uniexd  4535  abnexg  4541  snnex  4543  uniexb  4568  ssonuni  4584  dmexg  4994  rnexg  4995  elxp4  5222  elxp5  5223  iotaexab  5303  relrnfvex  5653  fvexg  5654  sefvex  5656  riotaexg  5970  iunexg  6276  1stvalg  6300  2ndvalg  6301  cnvf1o  6385  brtpos2  6412  tfrlemiex  6492  tfr1onlemex  6508  tfrcllemex  6521  en1bg  6969  en1uniel  6973  fival  7160  suplocexprlem2b  7924  suplocexprlemlub  7934  wrdexb  11115  restid  13323  tgval  13335  tgvalex  13336  istopon  14727  eltg  14766  eltg2  14767  tgss2  14793  ntrval  14824  restin  14890  cnovex  14910  cnprcl2k  14920  cnptopresti  14952  cnptoprest  14953  cnptoprest2  14954  lmtopcnp  14964  txbasex  14971  uptx  14988  reldvg  15393