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

Theorem trintssm 4042
 Description: Any inhabited transitive class includes its intersection. Similar to Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the inhabitedness hypothesis). (Contributed by Jim Kingdon, 22-Aug-2018.)
Assertion
Ref Expression
trintssm ((Tr 𝐴 ∧ ∃𝑥 𝑥𝐴) → 𝐴𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem trintssm
StepHypRef Expression
1 intss1 3786 . . . 4 (𝑥𝐴 𝐴𝑥)
2 trss 4035 . . . . 5 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
32com12 30 . . . 4 (𝑥𝐴 → (Tr 𝐴𝑥𝐴))
4 sstr2 3104 . . . 4 ( 𝐴𝑥 → (𝑥𝐴 𝐴𝐴))
51, 3, 4sylsyld 58 . . 3 (𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
65exlimiv 1577 . 2 (∃𝑥 𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
76impcom 124 1 ((Tr 𝐴 ∧ ∃𝑥 𝑥𝐴) → 𝐴𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∃wex 1468   ∈ wcel 1480   ⊆ wss 3071  ∩ cint 3771  Tr wtr 4026 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-int 3772  df-tr 4027 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator