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Theorem eluni 3743
 Description: Membership in class union. (Contributed by NM, 22-May-1994.)
Assertion
Ref Expression
eluni (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eluni
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elex 2698 . 2 (𝐴 𝐵𝐴 ∈ V)
2 elex 2698 . . . 4 (𝐴𝑥𝐴 ∈ V)
32adantr 274 . . 3 ((𝐴𝑥𝑥𝐵) → 𝐴 ∈ V)
43exlimiv 1578 . 2 (∃𝑥(𝐴𝑥𝑥𝐵) → 𝐴 ∈ V)
5 eleq1 2203 . . . . 5 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
65anbi1d 461 . . . 4 (𝑦 = 𝐴 → ((𝑦𝑥𝑥𝐵) ↔ (𝐴𝑥𝑥𝐵)))
76exbidv 1798 . . 3 (𝑦 = 𝐴 → (∃𝑥(𝑦𝑥𝑥𝐵) ↔ ∃𝑥(𝐴𝑥𝑥𝐵)))
8 df-uni 3741 . . 3 𝐵 = {𝑦 ∣ ∃𝑥(𝑦𝑥𝑥𝐵)}
97, 8elab2g 2832 . 2 (𝐴 ∈ V → (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵)))
101, 4, 9pm5.21nii 694 1 (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1332  ∃wex 1469   ∈ wcel 1481  Vcvv 2687  ∪ cuni 3740 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2689  df-uni 3741 This theorem is referenced by:  eluni2  3744  elunii  3745  eluniab  3752  uniun  3759  uniin  3760  uniss  3761  unissb  3770  dftr2  4032  unidif0  4095  unipw  4143  uniex2  4362  uniuni  4376  limom  4531  dmuni  4753  fununi  5195  nfvres  5458  elunirn  5671  tfrlem7  6218  tfrexlem  6235  tfrcldm  6264  fiuni  6870  isbasis2g  12242  tgval2  12250  ntreq0  12331  bj-uniex2  13268
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