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Theorem intss1 3756
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 2663 . . . 4 𝑥 ∈ V
21elint 3747 . . 3 (𝑥 𝐵 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
3 eleq1 2180 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
4 eleq2 2181 . . . . . 6 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
53, 4imbi12d 233 . . . . 5 (𝑦 = 𝐴 → ((𝑦𝐵𝑥𝑦) ↔ (𝐴𝐵𝑥𝐴)))
65spcgv 2747 . . . 4 (𝐴𝐵 → (∀𝑦(𝑦𝐵𝑥𝑦) → (𝐴𝐵𝑥𝐴)))
76pm2.43a 51 . . 3 (𝐴𝐵 → (∀𝑦(𝑦𝐵𝑥𝑦) → 𝑥𝐴))
82, 7syl5bi 151 . 2 (𝐴𝐵 → (𝑥 𝐵𝑥𝐴))
98ssrdv 3073 1 (𝐴𝐵 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1314   = wceq 1316  wcel 1465  wss 3041   cint 3741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-int 3742
This theorem is referenced by:  intminss  3766  intmin3  3768  intab  3770  int0el  3771  trintssm  4012  inteximm  4044  onnmin  4453  peano5  4482  peano5nnnn  7668  peano5nni  8691  dfuzi  9129  bj-intabssel  12923  bj-intabssel1  12924
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