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 43588
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 3967 . 2 (𝐴 ∈ (𝐵 {𝑥𝜑}) ↔ (𝐴𝐵𝐴 {𝑥𝜑}))
2 elintabg 4957 . . 3 (𝐴𝐵 → (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥)))
32pm5.32i 574 . 2 ((𝐴𝐵𝐴 {𝑥𝜑}) ↔ (𝐴𝐵 ∧ ∀𝑥(𝜑𝐴𝑥)))
41, 3bitri 275 1 (𝐴 ∈ (𝐵 {𝑥𝜑}) ↔ (𝐴𝐵 ∧ ∀𝑥(𝜑𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538  wcel 2108  {cab 2714  cin 3950   cint 4946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-v 3482  df-in 3958  df-int 4947
This theorem is referenced by:  inintabss  43591  inintabd  43592  elcnvcnvintab  43595
  Copyright terms: Public domain W3C validator