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

Theorem elinintab 37707
Description: Two ways of saying a set is an element of the intersection of a class with the intersection of a class. (Contributed by RP, 13-Aug-2020.)
Assertion
Ref Expression
elinintab (𝐴 ∈ (𝐵 {𝑥𝜑}) ↔ (𝐴𝐵 ∧ ∀𝑥(𝜑𝐴𝑥)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem elinintab
StepHypRef Expression
1 elin 3794 . 2 (𝐴 ∈ (𝐵 {𝑥𝜑}) ↔ (𝐴𝐵𝐴 {𝑥𝜑}))
2 elintabg 37706 . . 3 (𝐴𝐵 → (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥)))
32pm5.32i 669 . 2 ((𝐴𝐵𝐴 {𝑥𝜑}) ↔ (𝐴𝐵 ∧ ∀𝑥(𝜑𝐴𝑥)))
41, 3bitri 264 1 (𝐴 ∈ (𝐵 {𝑥𝜑}) ↔ (𝐴𝐵 ∧ ∀𝑥(𝜑𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1480  wcel 1989  {cab 2607  cin 3571   cint 4473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-v 3200  df-in 3579  df-int 4474
This theorem is referenced by:  inintabss  37710  inintabd  37711  elcnvcnvintab  37714
  Copyright terms: Public domain W3C validator