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Theorem trintss 5175
 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 4293 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 intss1 4877 . . . . 5 (𝑥𝐴 𝐴𝑥)
3 trss 5167 . . . . . 6 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
43com12 32 . . . . 5 (𝑥𝐴 → (Tr 𝐴𝑥𝐴))
5 sstr2 3960 . . . . 5 ( 𝐴𝑥 → (𝑥𝐴 𝐴𝐴))
62, 4, 5sylsyld 61 . . . 4 (𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
76exlimiv 1932 . . 3 (∃𝑥 𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
81, 7sylbi 220 . 2 (𝐴 ≠ ∅ → (Tr 𝐴 𝐴𝐴))
98impcom 411 1 ((Tr 𝐴𝐴 ≠ ∅) → 𝐴𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1781   ∈ wcel 2115   ≠ wne 3014   ⊆ wss 3919  ∅c0 4276  ∩ cint 4862  Tr wtr 5158 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-v 3482  df-dif 3922  df-in 3926  df-ss 3936  df-nul 4277  df-uni 4825  df-int 4863  df-tr 5159 This theorem is referenced by: (None)
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