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Theorem eliun 3631
Description: Membership in indexed union. (Contributed by NM, 3-Sep-2003.)
Assertion
Ref Expression
eliun (A x B 𝐶x B A 𝐶)
Distinct variable group:   x,A
Allowed substitution hints:   B(x)   𝐶(x)

Proof of Theorem eliun
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 elex 2539 . 2 (A x B 𝐶A V)
2 elex 2539 . . 3 (A 𝐶A V)
32rexlimivw 2403 . 2 (x B A 𝐶A V)
4 eleq1 2078 . . . 4 (y = A → (y 𝐶A 𝐶))
54rexbidv 2301 . . 3 (y = A → (x B y 𝐶x B A 𝐶))
6 df-iun 3629 . . 3 x B 𝐶 = {yx B y 𝐶}
75, 6elab2g 2662 . 2 (A V → (A x B 𝐶x B A 𝐶))
81, 3, 7pm5.21nii 607 1 (A x B 𝐶x B A 𝐶)
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1226   wcel 1370  wrex 2281  Vcvv 2531   ciun 3627
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-iun 3629
This theorem is referenced by:  iuncom  3633  iuncom4  3634  iunconstm  3635  iuniin  3637  iunss1  3638  ss2iun  3642  dfiun2g  3659  ssiun  3669  ssiun2  3670  iunab  3673  iun0  3683  0iun  3684  iunn0m  3687  iunin2  3690  iundif2ss  3692  iindif2m  3694  iunxsng  3702  iunun  3704  iunxun  3705  iunxiun  3706  iunpwss  3713  triun  3837  iunpw  4157  xpiundi  4321  xpiundir  4322  iunxpf  4407  cnvuni  4444  dmiun  4467  dmuni  4468  rniun  4657  dfco2  4743  dfco2a  4744  coiun  4753  fun11iun  5068  imaiun  5320  eluniimadm  5325  opabex3d  5667  opabex3  5668  smoiun  5834  tfrlemi14d  5864  tfrlemi14  5865
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