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Theorem clel5 3650
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.) Remove use of ax-10 2129, ax-11 2146, and ax-12 2166. (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 3220 . 2 (𝑋𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑋)
2 eqcom 2732 . . 3 (𝑥 = 𝑋𝑋 = 𝑥)
32rexbii 3083 . 2 (∃𝑥𝐴 𝑥 = 𝑋 ↔ ∃𝑥𝐴 𝑋 = 𝑥)
41, 3bitri 274 1 (𝑋𝐴 ↔ ∃𝑥𝐴 𝑋 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wcel 2098  wrex 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-cleq 2717  df-clel 2802  df-rex 3060
This theorem is referenced by:  dfss5  4263  iunid  5064  4fvwrd4  13654  wrdlen1  14538  phisum  16760  symgmov1  19350  disjunsn  32457  rp-abid  42919
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