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Theorem intss1 3604
Description: An element of a class includes the intersection of the class. Exercise 4 of [TakeutiZaring] p. 44 (with correction), generalized to classes. (Contributed by NM, 18-Nov-1995.)
Assertion
Ref Expression
intss1 (A B BA)

Proof of Theorem intss1
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2538 . . . 4 x V
21elint 3595 . . 3 (x By(y Bx y))
3 eleq1 2082 . . . . . 6 (y = A → (y BA B))
4 eleq2 2083 . . . . . 6 (y = A → (x yx A))
53, 4imbi12d 223 . . . . 5 (y = A → ((y Bx y) ↔ (A Bx A)))
65spcgv 2617 . . . 4 (A B → (y(y Bx y) → (A Bx A)))
76pm2.43a 45 . . 3 (A B → (y(y Bx y) → x A))
82, 7syl5bi 141 . 2 (A B → (x Bx A))
98ssrdv 2928 1 (A B BA)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1226   = wceq 1228   wcel 1374  wss 2894   cint 3589
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-in 2901  df-ss 2908  df-int 3590
This theorem is referenced by:  intminss  3614  intmin3  3616  intab  3618  int0el  3619  trint0m  3845  inteximm  3877  onnmin  4228  peano5  4248  bj-intabssel  7035  bj-intabssel1  7036  peano5set  7162
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