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Theorem intss1 3874
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 (𝐴𝐵 𝐵𝐴)

Proof of Theorem intss1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2755 . . . 4 𝑥 ∈ V
21elint 3865 . . 3 (𝑥 𝐵 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
3 eleq1 2252 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
4 eleq2 2253 . . . . . 6 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
53, 4imbi12d 234 . . . . 5 (𝑦 = 𝐴 → ((𝑦𝐵𝑥𝑦) ↔ (𝐴𝐵𝑥𝐴)))
65spcgv 2839 . . . 4 (𝐴𝐵 → (∀𝑦(𝑦𝐵𝑥𝑦) → (𝐴𝐵𝑥𝐴)))
76pm2.43a 51 . . 3 (𝐴𝐵 → (∀𝑦(𝑦𝐵𝑥𝑦) → 𝑥𝐴))
82, 7biimtrid 152 . 2 (𝐴𝐵 → (𝑥 𝐵𝑥𝐴))
98ssrdv 3176 1 (𝐴𝐵 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1362   = wceq 1364  wcel 2160  wss 3144   cint 3859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-int 3860
This theorem is referenced by:  intminss  3884  intmin3  3886  intab  3888  int0el  3889  trintssm  4132  inteximm  4167  onnmin  4585  peano5  4615  peano5nnnn  7922  peano5nni  8953  dfuzi  9394  bj-intabssel  15019  bj-intabssel1  15020
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