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Theorem intss1 3725
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 2636 . . . 4 𝑥 ∈ V
21elint 3716 . . 3 (𝑥 𝐵 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
3 eleq1 2157 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
4 eleq2 2158 . . . . . 6 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
53, 4imbi12d 233 . . . . 5 (𝑦 = 𝐴 → ((𝑦𝐵𝑥𝑦) ↔ (𝐴𝐵𝑥𝐴)))
65spcgv 2720 . . . 4 (𝐴𝐵 → (∀𝑦(𝑦𝐵𝑥𝑦) → (𝐴𝐵𝑥𝐴)))
76pm2.43a 51 . . 3 (𝐴𝐵 → (∀𝑦(𝑦𝐵𝑥𝑦) → 𝑥𝐴))
82, 7syl5bi 151 . 2 (𝐴𝐵 → (𝑥 𝐵𝑥𝐴))
98ssrdv 3045 1 (𝐴𝐵 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1294   = wceq 1296  wcel 1445  wss 3013   cint 3710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019  df-ss 3026  df-int 3711
This theorem is referenced by:  intminss  3735  intmin3  3737  intab  3739  int0el  3740  trintssm  3974  inteximm  4006  onnmin  4412  peano5  4441  peano5nnnn  7524  peano5nni  8523  dfuzi  8955  bj-intabssel  12397  bj-intabssel1  12398
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