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Theorem intss1 3658
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 2577 . . . 4 𝑥 ∈ V
21elint 3649 . . 3 (𝑥 𝐵 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
3 eleq1 2116 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
4 eleq2 2117 . . . . . 6 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
53, 4imbi12d 227 . . . . 5 (𝑦 = 𝐴 → ((𝑦𝐵𝑥𝑦) ↔ (𝐴𝐵𝑥𝐴)))
65spcgv 2657 . . . 4 (𝐴𝐵 → (∀𝑦(𝑦𝐵𝑥𝑦) → (𝐴𝐵𝑥𝐴)))
76pm2.43a 49 . . 3 (𝐴𝐵 → (∀𝑦(𝑦𝐵𝑥𝑦) → 𝑥𝐴))
82, 7syl5bi 145 . 2 (𝐴𝐵 → (𝑥 𝐵𝑥𝐴))
98ssrdv 2979 1 (𝐴𝐵 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257   = wceq 1259  wcel 1409  wss 2945   cint 3643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-int 3644
This theorem is referenced by:  intminss  3668  intmin3  3670  intab  3672  int0el  3673  trintssm  3898  inteximm  3931  onnmin  4320  peano5  4349  peano5nnnn  7024  peano5nni  7993  dfuzi  8407  bj-intabssel  10315  bj-intabssel1  10316  peano5setOLD  10452
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