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Theorem intss1 3688
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 2618 . . . 4 𝑥 ∈ V
21elint 3679 . . 3 (𝑥 𝐵 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
3 eleq1 2147 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
4 eleq2 2148 . . . . . 6 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
53, 4imbi12d 232 . . . . 5 (𝑦 = 𝐴 → ((𝑦𝐵𝑥𝑦) ↔ (𝐴𝐵𝑥𝐴)))
65spcgv 2699 . . . 4 (𝐴𝐵 → (∀𝑦(𝑦𝐵𝑥𝑦) → (𝐴𝐵𝑥𝐴)))
76pm2.43a 50 . . 3 (𝐴𝐵 → (∀𝑦(𝑦𝐵𝑥𝑦) → 𝑥𝐴))
82, 7syl5bi 150 . 2 (𝐴𝐵 → (𝑥 𝐵𝑥𝐴))
98ssrdv 3020 1 (𝐴𝐵 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1285   = wceq 1287  wcel 1436  wss 2988   cint 3673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-in 2994  df-ss 3001  df-int 3674
This theorem is referenced by:  intminss  3698  intmin3  3700  intab  3702  int0el  3703  trintssm  3929  inteximm  3962  onnmin  4359  peano5  4388  peano5nnnn  7374  peano5nni  8363  dfuzi  8792  bj-intabssel  11158  bj-intabssel1  11159
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