MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  trintss Structured version   Visualization version   GIF version

Theorem trintss 5245
Description: Any nonempty transitive class includes its intersection. Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the nonemptiness hypothesis). (Contributed by Scott Fenton, 3-Mar-2011.) (Proof shortened by Andrew Salmon, 14-Nov-2011.)
Assertion
Ref Expression
trintss ((Tr 𝐴𝐴 ≠ ∅) → 𝐴𝐴)

Proof of Theorem trintss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 n0 4310 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 intss1 4928 . . . . 5 (𝑥𝐴 𝐴𝑥)
3 trss 5237 . . . . . 6 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
43com12 32 . . . . 5 (𝑥𝐴 → (Tr 𝐴𝑥𝐴))
5 sstr2 3955 . . . . 5 ( 𝐴𝑥 → (𝑥𝐴 𝐴𝐴))
62, 4, 5sylsyld 61 . . . 4 (𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
76exlimiv 1934 . . 3 (∃𝑥 𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
81, 7sylbi 216 . 2 (𝐴 ≠ ∅ → (Tr 𝐴 𝐴𝐴))
98impcom 409 1 ((Tr 𝐴𝐴 ≠ ∅) → 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wex 1782  wcel 2107  wne 2940  wss 3914  c0 4286   cint 4911  Tr wtr 5226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-v 3449  df-dif 3917  df-in 3921  df-ss 3931  df-nul 4287  df-uni 4870  df-int 4912  df-tr 5227
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator