Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elintabg Structured version   Visualization version   GIF version

Theorem elintabg 36796
Description: Two ways of saying a set is an element of the intersection of a class. (Contributed by RP, 13-Aug-2020.)
Assertion
Ref Expression
elintabg (𝐴𝑉 → (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem elintabg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elintg 4316 . 2 (𝐴𝑉 → (𝐴 {𝑥𝜑} ↔ ∀𝑦 ∈ {𝑥𝜑}𝐴𝑦))
2 eleq2 2581 . . 3 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
32ralab2 3242 . 2 (∀𝑦 ∈ {𝑥𝜑}𝐴𝑦 ↔ ∀𝑥(𝜑𝐴𝑥))
41, 3syl6bb 274 1 (𝐴𝑉 → (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472  wcel 1938  {cab 2500  wral 2800   cint 4308
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-v 3079  df-int 4309
This theorem is referenced by:  elinintab  36797
  Copyright terms: Public domain W3C validator