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Theorem risset 2440
Description: Two ways to say "𝐴 belongs to 𝐵." (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
risset (𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem risset
StepHypRef Expression
1 exancom 1572 . 2 (∃𝑥(𝑥𝐵𝑥 = 𝐴) ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
2 df-rex 2399 . 2 (∃𝑥𝐵 𝑥 = 𝐴 ↔ ∃𝑥(𝑥𝐵𝑥 = 𝐴))
3 df-clel 2113 . 2 (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
41, 2, 33bitr4ri 212 1 (𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1316  wex 1453  wcel 1465  wrex 2394
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-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-clel 2113  df-rex 2399
This theorem is referenced by:  reueq  2856  reuind  2862  0el  3355  iunid  3838  sucel  4302  reusv3  4351  fvmptt  5480  releldm2  6051  qsid  6462  rerecclap  8458  nndiv  8729  zq  9386  4fvwrd4  9885  bj-bdcel  12962
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