Theorem eluni 4862

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 3457	. 2 ⊢ (𝐴 ∈ ∪ 𝐵 → 𝐴 ∈ V)
2		elex 3457	. . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ V)
3	2	adantr 480	. . 3 ⊢ ((𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ V)
4	3	exlimiv 1931	. 2 ⊢ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ V)
5		eleq1 2819	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
6	5	anbi1d 631	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)))
7	6	exbidv 1922	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)))
8		df-uni 4860	. . 3 ⊢ ∪ 𝐵 = {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)}
9	7, 8	elab2g 3636	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)))
10	1, 4, 9	pm5.21nii 378	1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  Vcvv 3436  ∪ cuni 4859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-uni 4860

This theorem is referenced by:  eluni2  4863  elunii  4864  uniss  4867  eluniab  4873  uniun  4882  uniin  4883  unissb  4891  dfiun2g  4980  dftr2  5200  unipw  5391  dmuni  5854  iotanul2  6454  fununi  6556  elunirn  7185  uniex2  7671  uniuni  7695  mpoxopxnop0  8145  fprresex  8240  tfrlem7  8302  tfrlem9a  8305  inf2  9513  inf3lem2  9519  rankwflemb  9683  cardprclem  9869  carduni  9871  iunfictbso  10002  kmlem3  10041  kmlem4  10042  cfub  10137  isf34lem4  10265  grothtsk  10723  suplem1pr  10940  toprntopon  22838  isbasis2g  22861  tgval2  22869  ntreq0  22990  cmpsublem  23312  cmpsub  23313  cmpcld  23315  is1stc2  23355  alexsubALTlem3  23962  alexsubALT  23964  elold  27812  fnessref  36390  bj-restuni  37130  difunieq  37407  ismnushort  44333  truniALT  44573  truniALTVD  44909  unisnALT  44957  uniclaxun  45018  elunif  45052  ssfiunibd  45349  stoweidlem27  46064  stoweidlem48  46085  setrec1lem3  49720  setrec1  49722