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Theorem elun 3057
Description: Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.)
Assertion
Ref Expression
elun (A (B𝐶) ↔ (A B A 𝐶))

Proof of Theorem elun
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elex 2539 . 2 (A (B𝐶) → A V)
2 elex 2539 . . 3 (A BA V)
3 elex 2539 . . 3 (A 𝐶A V)
42, 3jaoi 623 . 2 ((A B A 𝐶) → A V)
5 eleq1 2078 . . . 4 (x = A → (x BA B))
6 eleq1 2078 . . . 4 (x = A → (x 𝐶A 𝐶))
75, 6orbi12d 694 . . 3 (x = A → ((x B x 𝐶) ↔ (A B A 𝐶)))
8 df-un 2895 . . 3 (B𝐶) = {x ∣ (x B x 𝐶)}
97, 8elab2g 2662 . 2 (A V → (A (B𝐶) ↔ (A B A 𝐶)))
101, 4, 9pm5.21nii 607 1 (A (B𝐶) ↔ (A B A 𝐶))
Colors of variables: wff set class
Syntax hints:  wb 98   wo 616   = wceq 1226   wcel 1370  Vcvv 2531  cun 2888
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-v 2533  df-un 2895
This theorem is referenced by:  uneqri  3058  uncom  3060  uneq1  3063  unass  3073  ssun1  3079  unss1  3085  ssequn1  3086  unss  3090  rexun  3096  ralunb  3097  unssdif  3145  unssin  3149  inssun  3150  indi  3157  undi  3158  difundi  3162  difindiss  3164  undif3ss  3171  symdifxor  3176  rabun2  3189  reuun2  3193  undif4  3257  ssundifim  3279  dfpr2  3362  eltpg  3386  pwprss  3546  pwtpss  3547  uniun  3569  intun  3616  iunun  3704  iunxun  3705  iinuniss  3707  brun  3780  pwunss  3990  elsuci  4085  elsucg  4086  elsuc2g  4087  ordsucim  4172  sucprcreg  4207  opthprc  4314  xpundi  4319  xpundir  4320  funun  4866  mptun  4951  unpreima  5213  reldmtpos  5786  dftpos4  5796  tpostpos  5797  bj-nntrans  7312  bj-nnelirr  7314
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