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Theorem clel5 40303
 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.)
Assertion
Ref Expression
clel5 (𝑋𝐴 ↔ ∃𝑥𝐴 𝑋 = 𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋

Proof of Theorem clel5
StepHypRef Expression
1 id 22 . . 3 (𝑋𝐴𝑋𝐴)
2 eqeq2 2621 . . . 4 (𝑥 = 𝑋 → (𝑋 = 𝑥𝑋 = 𝑋))
32adantl 481 . . 3 ((𝑋𝐴𝑥 = 𝑋) → (𝑋 = 𝑥𝑋 = 𝑋))
4 eqidd 2611 . . 3 (𝑋𝐴𝑋 = 𝑋)
51, 3, 4rspcedvd 3289 . 2 (𝑋𝐴 → ∃𝑥𝐴 𝑋 = 𝑥)
6 eleq1a 2683 . . 3 (𝑥𝐴 → (𝑋 = 𝑥𝑋𝐴))
76rexlimiv 3009 . 2 (∃𝑥𝐴 𝑋 = 𝑥𝑋𝐴)
85, 7impbii 198 1 (𝑋𝐴 ↔ ∃𝑥𝐴 𝑋 = 𝑥)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175 This theorem is referenced by:  dfss7  40304
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