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Theorem eluni 3574
Description: Membership in class union. (Contributed by NM, 22-May-1994.)
Assertion
Ref Expression
eluni (A Bx(A x x B))
Distinct variable groups:   x,A   x,B

Proof of Theorem eluni
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 elex 2560 . 2 (A BA V)
2 elex 2560 . . . 4 (A xA V)
32adantr 261 . . 3 ((A x x B) → A V)
43exlimiv 1486 . 2 (x(A x x B) → A V)
5 eleq1 2097 . . . . 5 (y = A → (y xA x))
65anbi1d 438 . . . 4 (y = A → ((y x x B) ↔ (A x x B)))
76exbidv 1703 . . 3 (y = A → (x(y x x B) ↔ x(A x x B)))
8 df-uni 3572 . . 3 B = {yx(y x x B)}
97, 8elab2g 2683 . 2 (A V → (A Bx(A x x B)))
101, 4, 9pm5.21nii 619 1 (A Bx(A x x B))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551   cuni 3571
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-uni 3572
This theorem is referenced by:  eluni2  3575  elunii  3576  eluniab  3583  uniun  3590  uniin  3591  uniss  3592  unissb  3601  dftr2  3847  unidif0  3911  unipw  3944  uniex2  4139  uniuni  4149  limom  4279  dmuni  4488  fununi  4910  nfvres  5149  elunirn  5348  tfrlem7  5874  tfrexlem  5889  bj-uniex2  9301
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