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Theorem trintss 4919
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 4072 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 intss1 4642 . . . . 5 (𝑥𝐴 𝐴𝑥)
3 trss 4911 . . . . . 6 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
43com12 32 . . . . 5 (𝑥𝐴 → (Tr 𝐴𝑥𝐴))
5 sstr2 3749 . . . . 5 ( 𝐴𝑥 → (𝑥𝐴 𝐴𝐴))
62, 4, 5sylsyld 61 . . . 4 (𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
76exlimiv 2005 . . 3 (∃𝑥 𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
81, 7sylbi 207 . 2 (𝐴 ≠ ∅ → (Tr 𝐴 𝐴𝐴))
98impcom 445 1 ((Tr 𝐴𝐴 ≠ ∅) → 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wex 1851  wcel 2137  wne 2930  wss 3713  c0 4056   cint 4625  Tr wtr 4902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-v 3340  df-dif 3716  df-in 3720  df-ss 3727  df-nul 4057  df-uni 4587  df-int 4626  df-tr 4903
This theorem is referenced by: (None)
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