Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  trint0m GIF version

Theorem trint0m 3862
 Description: Any inhabited transitive class includes its intersection. Similar to Exercise 2 in [TakeutiZaring] p. 44. (Contributed by Jim Kingdon, 22-Aug-2018.)
Assertion
Ref Expression
trint0m ((Tr A x x A) → AA)
Distinct variable group:   x,A

Proof of Theorem trint0m
StepHypRef Expression
1 intss1 3621 . . . 4 (x A Ax)
2 trss 3854 . . . . 5 (Tr A → (x AxA))
32com12 27 . . . 4 (x A → (Tr AxA))
4 sstr2 2946 . . . 4 ( Ax → (xA AA))
51, 3, 4sylsyld 52 . . 3 (x A → (Tr A AA))
65exlimiv 1486 . 2 (x x A → (Tr A AA))
76impcom 116 1 ((Tr A x x A) → AA)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1378   ∈ wcel 1390   ⊆ wss 2911  ∩ cint 3606  Tr wtr 3845 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-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 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-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-int 3607  df-tr 3846 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator