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Theorem clel5 3484
 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 2772 . . . 4 (𝑥 = 𝑋 → (𝑋 = 𝑥𝑋 = 𝑋))
32adantl 473 . . 3 ((𝑋𝐴𝑥 = 𝑋) → (𝑋 = 𝑥𝑋 = 𝑋))
4 eqidd 2762 . . 3 (𝑋𝐴𝑋 = 𝑋)
51, 3, 4rspcedvd 3457 . 2 (𝑋𝐴 → ∃𝑥𝐴 𝑋 = 𝑥)
6 eleq1a 2835 . . 3 (𝑥𝐴 → (𝑋 = 𝑥𝑋𝐴))
76rexlimiv 3166 . 2 (∃𝑥𝐴 𝑋 = 𝑥𝑋𝐴)
85, 7impbii 199 1 (𝑋𝐴 ↔ ∃𝑥𝐴 𝑋 = 𝑥)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1632   ∈ wcel 2140  ∃wrex 3052 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-v 3343 This theorem is referenced by:  dfss5  4008  disjunsn  29736
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