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Theorem elunii 3576
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 2098 . . . . 5 (x = B → (A xA B))
2 eleq1 2097 . . . . 5 (x = B → (x 𝐶B 𝐶))
31, 2anbi12d 442 . . . 4 (x = B → ((A x x 𝐶) ↔ (A B B 𝐶)))
43spcegv 2635 . . 3 (B 𝐶 → ((A B B 𝐶) → x(A x x 𝐶)))
54anabsi7 515 . 2 ((A B B 𝐶) → x(A x x 𝐶))
6 eluni 3574 . 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 1242  wex 1378   wcel 1390   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:  ssuni  3593  unipw  3944  opeluu  4148  sucunielr  4201  unon  4202  ordunisuc2r  4205  tfrlemibxssdm  5882
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