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Theorem eluni 3583
 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 2566 . 2 (𝐴 𝐵𝐴 ∈ V)
2 elex 2566 . . . 4 (𝐴𝑥𝐴 ∈ V)
32adantr 261 . . 3 ((𝐴𝑥𝑥𝐵) → 𝐴 ∈ V)
43exlimiv 1489 . 2 (∃𝑥(𝐴𝑥𝑥𝐵) → 𝐴 ∈ V)
5 eleq1 2100 . . . . 5 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
65anbi1d 438 . . . 4 (𝑦 = 𝐴 → ((𝑦𝑥𝑥𝐵) ↔ (𝐴𝑥𝑥𝐵)))
76exbidv 1706 . . 3 (𝑦 = 𝐴 → (∃𝑥(𝑦𝑥𝑥𝐵) ↔ ∃𝑥(𝐴𝑥𝑥𝐵)))
8 df-uni 3581 . . 3 𝐵 = {𝑦 ∣ ∃𝑥(𝑦𝑥𝑥𝐵)}
97, 8elab2g 2689 . 2 (𝐴 ∈ V → (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵)))
101, 4, 9pm5.21nii 620 1 (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557  ∪ cuni 3580 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-uni 3581 This theorem is referenced by:  eluni2  3584  elunii  3585  eluniab  3592  uniun  3599  uniin  3600  uniss  3601  unissb  3610  dftr2  3856  unidif0  3920  unipw  3953  uniex2  4173  uniuni  4183  limom  4336  dmuni  4545  fununi  4967  nfvres  5206  elunirn  5405  tfrlem7  5933  tfrexlem  5948  bj-uniex2  10036
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