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Theorem trintss 5275
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 4339 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 intss1 4958 . . . . 5 (𝑥𝐴 𝐴𝑥)
3 trss 5267 . . . . . 6 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
43com12 32 . . . . 5 (𝑥𝐴 → (Tr 𝐴𝑥𝐴))
5 sstr2 3982 . . . . 5 ( 𝐴𝑥 → (𝑥𝐴 𝐴𝐴))
62, 4, 5sylsyld 61 . . . 4 (𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
76exlimiv 1925 . . 3 (∃𝑥 𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
81, 7sylbi 216 . 2 (𝐴 ≠ ∅ → (Tr 𝐴 𝐴𝐴))
98impcom 407 1 ((Tr 𝐴𝐴 ≠ ∅) → 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1773  wcel 2098  wne 2932  wss 3941  c0 4315   cint 4941  Tr wtr 5256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-v 3468  df-dif 3944  df-in 3948  df-ss 3958  df-nul 4316  df-uni 4901  df-int 4942  df-tr 5257
This theorem is referenced by: (None)
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