Theorem eluni 4873

Description: Membership in class union. (Contributed by NM, 22-May-1994.)

Assertion

Ref	Expression
eluni	⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eluni

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3475	. 2 ⊢ (𝐴 ∈ ∪ 𝐵 → 𝐴 ∈ V)
2		elex 3475	. . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ V)
3	2	adantr 486	. . 3 ⊢ ((𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ V)
4	3	exlimiv 1965	. 2 ⊢ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ V)
5		eleq1 2852	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
6	5	anbi1d 643	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)))
7	6	exbidv 1956	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)))
8		df-uni 4871	. . 3 ⊢ ∪ 𝐵 = {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)}
9	7, 8	elab2g 3637	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)))
10	1, 4, 9	pm5.21nii 381	1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454  ∪ cuni 4870

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-uni 4871

This theorem is used by:  eluni2  4874  elunii  4875  uniss  4878  eluniab  4884  uniun  4893  uniinOLD  4895  uni0  4899  unissb  4904  dfiun2g  4991  dftr2  5218  unipw  5423  dmuni  5904  iotanul2  6532  fununi  6640  elunirn  7281  uniex2  7781  uniex2OLD  7782  uniuni  7803  mpoxopxnop0  8263  fprresex  8359  tfrlem7  8422  tfrlem9a  8425  inf2  9654  inf3lem2  9660  rankwflemb  9830  rankwflembOLD  9831  setrec1lem3  9999  setrec1  10002  cardprclem  10090  carduni  10092  iunfictbso  10223  kmlem3  10261  kmlem4  10262  cfub  10356  isf34lem4  10485  grothtsk  10950  suplem1pr  11167  lidlunin0  21548  toprntopon  23276  isbasis2g  23299  tgval2  23307  ntreq0  23428  cmpsublem  23750  cmpsub  23751  cmpcld  23753  is1stc2  23793  alexsubALTlem3  24401  alexsubALT  24403  elold  28278  fnessref  37336  mh-infprim1bi  37525  bj-restuni  38222  difunieq  38502  ismnushort  45488  truniALT  45726  truniALTVD  46062  unisnALT  46110  uniclaxun  46182  elunif  46249  ssfiunibd  46540  stoweidlem27  47252  stoweidlem48  47273