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

Theorem rintm 3912
Description: Relative intersection of an inhabited class. (Contributed by Jim Kingdon, 19-Aug-2018.)
Assertion
Ref Expression
rintm ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → (𝐴 𝑋) = 𝑋)
Distinct variable group:   𝑥,𝑋
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem rintm
StepHypRef Expression
1 incom 3272 . 2 (𝐴 𝑋) = ( 𝑋𝐴)
2 intssuni2m 3802 . . . 4 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → 𝑋 𝒫 𝐴)
3 ssid 3121 . . . . 5 𝒫 𝐴 ⊆ 𝒫 𝐴
4 sspwuni 3904 . . . . 5 (𝒫 𝐴 ⊆ 𝒫 𝐴 𝒫 𝐴𝐴)
53, 4mpbi 144 . . . 4 𝒫 𝐴𝐴
62, 5sstrdi 3113 . . 3 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → 𝑋𝐴)
7 df-ss 3088 . . 3 ( 𝑋𝐴 ↔ ( 𝑋𝐴) = 𝑋)
86, 7sylib 121 . 2 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → ( 𝑋𝐴) = 𝑋)
91, 8syl5eq 2185 1 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → (𝐴 𝑋) = 𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wex 1469  wcel 1481  cin 3074  wss 3075  𝒫 cpw 3514   cuni 3743   cint 3778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3081  df-ss 3088  df-pw 3516  df-uni 3744  df-int 3779
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator