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

Theorem intid 3951
 Description: The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.)
Hypothesis
Ref Expression
intid.1 A V
Assertion
Ref Expression
intid {xA x} = {A}
Distinct variable group:   x,A

Proof of Theorem intid
StepHypRef Expression
1 intid.1 . . . 4 A V
2 snexgOLD 3926 . . . 4 (A V → {A} V)
31, 2ax-mp 7 . . 3 {A} V
4 eleq2 2098 . . . 4 (x = {A} → (A xA {A}))
51snid 3394 . . . 4 A {A}
64, 5intmin3 3633 . . 3 ({A} V → {xA x} ⊆ {A})
73, 6ax-mp 7 . 2 {xA x} ⊆ {A}
81elintab 3617 . . . 4 (A {xA x} ↔ x(A xA x))
9 id 19 . . . 4 (A xA x)
108, 9mpgbir 1339 . . 3 A {xA x}
11 snssi 3499 . . 3 (A {xA x} → {A} ⊆ {xA x})
1210, 11ax-mp 7 . 2 {A} ⊆ {xA x}
137, 12eqssi 2955 1 {xA x} = {A}
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  {cab 2023  Vcvv 2551   ⊆ wss 2911  {csn 3367  ∩ cint 3606 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-int 3607 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator