MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clel5 Structured version   Visualization version   GIF version

Theorem clel5 3375
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 2662 . . . 4 (𝑥 = 𝑋 → (𝑋 = 𝑥𝑋 = 𝑋))
32adantl 481 . . 3 ((𝑋𝐴𝑥 = 𝑋) → (𝑋 = 𝑥𝑋 = 𝑋))
4 eqidd 2652 . . 3 (𝑋𝐴𝑋 = 𝑋)
51, 3, 4rspcedvd 3348 . 2 (𝑋𝐴 → ∃𝑥𝐴 𝑋 = 𝑥)
6 eleq1a 2725 . . 3 (𝑥𝐴 → (𝑋 = 𝑥𝑋𝐴))
76rexlimiv 3056 . 2 (∃𝑥𝐴 𝑋 = 𝑥𝑋𝐴)
85, 7impbii 199 1 (𝑋𝐴 ↔ ∃𝑥𝐴 𝑋 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  wcel 2030  wrex 2942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233
This theorem is referenced by:  dfss5  3897  disjunsn  29533
  Copyright terms: Public domain W3C validator