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Theorem clel5 2861
Description: Alternate definition of class membership: a class 𝑋 is an element of another class 𝐴 iff there is an element of 𝐴 equal to 𝑋. (Contributed by AV, 13-Nov-2020.) (Revised by Steven Nguyen, 19-May-2023.)
Assertion
Ref Expression
clel5 (𝑋𝐴 ↔ ∃𝑥𝐴 𝑋 = 𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋

Proof of Theorem clel5
StepHypRef Expression
1 risset 2492 . 2 (𝑋𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑋)
2 eqcom 2166 . . 3 (𝑥 = 𝑋𝑋 = 𝑥)
32rexbii 2471 . 2 (∃𝑥𝐴 𝑥 = 𝑋 ↔ ∃𝑥𝐴 𝑋 = 𝑥)
41, 3bitri 183 1 (𝑋𝐴 ↔ ∃𝑥𝐴 𝑋 = 𝑥)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1342  wcel 2135  wrex 2443
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 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-17 1513  ax-ial 1521  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-cleq 2157  df-clel 2160  df-rex 2448
This theorem is referenced by:  phisum  12166
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