Theorem eluni 4880

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 3479	. 2 ⊢ (𝐴 ∈ ∪ 𝐵 → 𝐴 ∈ V)
2		elex 3479	. . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ V)
3	2	adantr 486	. . 3 ⊢ ((𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ V)
4	3	exlimiv 1963	. 2 ⊢ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ V)
5		eleq1 2854	. . . . 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 1954	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)))
8		df-uni 4878	. . 3 ⊢ ∪ 𝐵 = {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)}
9	7, 8	elab2g 3642	. 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 2146  Vcvv 3458  ∪ cuni 4877

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-uni 4878

This theorem is used by:  eluni2  4881  elunii  4882  uniss  4885  eluniab  4891  uniun  4900  uniinOLD  4902  uni0  4906  unissb  4911  dfiun2g  4999  dftr2  5225  unipw  5436  dmuni  5909  iotanul2  6516  fununi  6618  elunirn  7256  uniex2  7748  uniex2OLD  7749  uniuni  7770  mpoxopxnop0  8220  fprresex  8316  tfrlem7  8379  tfrlem9a  8382  inf2  9602  inf3lem2  9608  rankwflemb  9775  cardprclem  9984  carduni  9986  iunfictbso  10117  kmlem3  10155  kmlem4  10156  cfub  10250  isf34lem4  10379  grothtsk  10838  suplem1pr  11055  lidlunin0  21398  toprntopon  23119  isbasis2g  23142  tgval2  23150  ntreq0  23271  cmpsublem  23593  cmpsub  23594  cmpcld  23596  is1stc2  23636  alexsubALTlem3  24243  alexsubALT  24245  elold  28089  fnessref  36909  mh-infprim1bi  37098  bj-restuni  37780  difunieq  38061  ismnushort  45052  truniALT  45291  truniALTVD  45627  unisnALT  45675  uniclaxun  45736  elunif  45777  ssfiunibd  46069  stoweidlem27  46782  stoweidlem48  46803  setrec1lem3  50508  setrec1  50510