Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  intss1 Structured version   GIF version

Theorem intss1 3620
 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 2554 . . . 4 x V
21elint 3611 . . 3 (x By(y Bx y))
3 eleq1 2097 . . . . . 6 (y = A → (y BA B))
4 eleq2 2098 . . . . . 6 (y = A → (x yx A))
53, 4imbi12d 223 . . . . 5 (y = A → ((y Bx y) ↔ (A Bx A)))
65spcgv 2634 . . . 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 2945 1 (A B BA)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1240   = wceq 1242   ∈ wcel 1390   ⊆ wss 2911  ∩ cint 3605 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-in 2918  df-ss 2925  df-int 3606 This theorem is referenced by:  intminss  3630  intmin3  3632  intab  3634  int0el  3635  trint0m  3861  inteximm  3893  onnmin  4243  peano5  4263  peano5nni  7649  dfuzi  8073  bj-intabssel  8862  bj-intabssel1  8863  peano5set  8993
 Copyright terms: Public domain W3C validator