Theorem eluni 4876

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 3476	. 2 ⊢ (𝐴 ∈ ∪ 𝐵 → 𝐴 ∈ V)
2		elex 3476	. . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ V)
3	2	adantr 485	. . 3 ⊢ ((𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ V)
4	3	exlimiv 1960	. 2 ⊢ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ V)
5		eleq1 2851	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
6	5	anbi1d 642	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)))
7	6	exbidv 1951	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)))
8		df-uni 4874	. . 3 ⊢ ∪ 𝐵 = {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)}
9	7, 8	elab2g 3640	. 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))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455  ∪ cuni 4873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-uni 4874

This theorem is referenced by:  eluni2  4877  elunii  4878  uniss  4881  eluniab  4887  uniun  4896  uniinOLD  4898  uni0  4902  unissb  4907  dfiun2g  4995  dftr2  5221  unipw  5433  dmuni  5906  iotanul2  6511  fununi  6613  elunirn  7251  uniex2  7737  uniex2OLD  7738  uniuni  7762  mpoxopxnop0  8212  fprresex  8308  tfrlem7  8371  tfrlem9a  8374  inf2  9593  inf3lem2  9599  rankwflemb  9766  cardprclem  9966  carduni  9968  iunfictbso  10099  kmlem3  10137  kmlem4  10138  cfub  10233  isf34lem4  10362  grothtsk  10821  suplem1pr  11038  lidlunin0  21342  toprntopon  23063  isbasis2g  23086  tgval2  23094  ntreq0  23215  cmpsublem  23537  cmpsub  23538  cmpcld  23540  is1stc2  23580  alexsubALTlem3  24187  alexsubALT  24189  elold  28033  fnessref  36849  mh-infprim1bi  37038  bj-restuni  37720  difunieq  38001  ismnushort  44994  truniALT  45233  truniALTVD  45569  unisnALT  45617  uniclaxun  45678  elunif  45719  ssfiunibd  46011  stoweidlem27  46724  stoweidlem48  46745  setrec1lem3  50450  setrec1  50452