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Theorem trint0m 3845
 Description: Any inhabited transitive class includes its intersection. Similar to Exercise 2 in [TakeutiZaring] p. 44. (Contributed by Jim Kingdon, 22-Aug-2018.)
Assertion
Ref Expression
trint0m ((Tr A x x A) → AA)
Distinct variable group:   x,A

Proof of Theorem trint0m
StepHypRef Expression
1 intss1 3604 . . . 4 (x A Ax)
2 trss 3837 . . . . 5 (Tr A → (x AxA))
32com12 27 . . . 4 (x A → (Tr AxA))
4 sstr2 2929 . . . 4 ( Ax → (xA AA))
51, 3, 4sylsyld 52 . . 3 (x A → (Tr A AA))
65exlimiv 1471 . 2 (x x A → (Tr A AA))
76impcom 116 1 ((Tr A x x A) → AA)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1362   ∈ wcel 1374   ⊆ wss 2894  ∩ cint 3589  Tr wtr 3828 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-in 2901  df-ss 2908  df-uni 3555  df-int 3590  df-tr 3829 This theorem is referenced by: (None)
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