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Theorem elunii 3559
Description: Membership in class union. (Contributed by NM, 24-Mar-1995.)
Assertion
Ref Expression
elunii ((A B B 𝐶) → A 𝐶)

Proof of Theorem elunii
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2083 . . . . 5 (x = B → (A xA B))
2 eleq1 2082 . . . . 5 (x = B → (x 𝐶B 𝐶))
31, 2anbi12d 445 . . . 4 (x = B → ((A x x 𝐶) ↔ (A B B 𝐶)))
43spcegv 2618 . . 3 (B 𝐶 → ((A B B 𝐶) → x(A x x 𝐶)))
54anabsi7 502 . 2 ((A B B 𝐶) → x(A x x 𝐶))
6 eluni 3557 . 2 (A 𝐶x(A x x 𝐶))
75, 6sylibr 137 1 ((A B B 𝐶) → A 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228  wex 1362   wcel 1374   cuni 3554
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-uni 3555
This theorem is referenced by:  ssuni  3576  unipw  3927  opeluu  4132  sucunielr  4185  unon  4186  ordunisuc2r  4189  tfrlemibxssdm  5862
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