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Theorem clel5 40211
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 2525 . . . 4 (𝑥 = 𝑋 → (𝑋 = 𝑥𝑋 = 𝑋))
32adantl 480 . . 3 ((𝑋𝐴𝑥 = 𝑋) → (𝑋 = 𝑥𝑋 = 𝑋))
4 eqidd 2515 . . 3 (𝑋𝐴𝑋 = 𝑋)
51, 3, 4rspcedvd 3193 . 2 (𝑋𝐴 → ∃𝑥𝐴 𝑋 = 𝑥)
6 eleq1a 2587 . . 3 (𝑥𝐴 → (𝑋 = 𝑥𝑋𝐴))
76rexlimiv 2913 . 2 (∃𝑥𝐴 𝑋 = 𝑥𝑋𝐴)
85, 7impbii 197 1 (𝑋𝐴 ↔ ∃𝑥𝐴 𝑋 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  wcel 1938  wrex 2801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-v 3079
This theorem is referenced by:  dfss7  40212
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