Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eliin2 Structured version   Visualization version   GIF version

Theorem eliin2 38330
 Description: Membership in indexed intersection. See eliincex 38324 for a counterexample showing that the precondition 𝐵 ≠ ∅ cannot be simply dropped. eliin 4461 uses an alternative precondition (and it doesn't have a disjoint var constraint between 𝐵 and 𝑥; see eliin2f 38316). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Assertion
Ref Expression
eliin2 (𝐵 ≠ ∅ → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem eliin2
StepHypRef Expression
1 nfcv 2751 . 2 𝑥𝐵
21eliin2f 38316 1 (𝐵 ≠ ∅ → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∅c0 3874  ∩ ciin 4456 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-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-nul 3875  df-iin 4458 This theorem is referenced by:  eliuniin2  38335  allbutfi  38557
 Copyright terms: Public domain W3C validator