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Theorem trintss 5284
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 4359 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 intss1 4968 . . . . 5 (𝑥𝐴 𝐴𝑥)
3 trss 5276 . . . . . 6 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
43com12 32 . . . . 5 (𝑥𝐴 → (Tr 𝐴𝑥𝐴))
5 sstr2 4002 . . . . 5 ( 𝐴𝑥 → (𝑥𝐴 𝐴𝐴))
62, 4, 5sylsyld 61 . . . 4 (𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
76exlimiv 1928 . . 3 (∃𝑥 𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
81, 7sylbi 217 . 2 (𝐴 ≠ ∅ → (Tr 𝐴 𝐴𝐴))
98impcom 407 1 ((Tr 𝐴𝐴 ≠ ∅) → 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1776  wcel 2106  wne 2938  wss 3963  c0 4339   cint 4951  Tr wtr 5265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-v 3480  df-dif 3966  df-ss 3980  df-nul 4340  df-uni 4913  df-int 4952  df-tr 5266
This theorem is referenced by: (None)
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