ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  intexrabim GIF version

Theorem intexrabim 3957
Description: The intersection of an inhabited restricted class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
Assertion
Ref Expression
intexrabim (∃𝑥𝐴 𝜑 {𝑥𝐴𝜑} ∈ V)

Proof of Theorem intexrabim
StepHypRef Expression
1 intexabim 3956 . 2 (∃𝑥(𝑥𝐴𝜑) → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
2 df-rex 2361 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
3 df-rab 2364 . . . 4 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
43inteqi 3669 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
54eleq1i 2150 . 2 ( {𝑥𝐴𝜑} ∈ V ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
61, 2, 53imtr4i 199 1 (∃𝑥𝐴 𝜑 {𝑥𝐴𝜑} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wex 1424  wcel 1436  {cab 2071  wrex 2356  {crab 2359  Vcvv 2614   cint 3665
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2616  df-in 2992  df-ss 2999  df-int 3666
This theorem is referenced by:  cardcl  6730  isnumi  6731  cardval3ex  6734
  Copyright terms: Public domain W3C validator