Theorem trintssm 3891

Description: Any inhabited transitive class includes its intersection. Similar to Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the inhabitedness hypothesis). (Contributed by Jim Kingdon, 22-Aug-2018.)

Assertion

Ref	Expression
trintssm	|- ( ( Tr A /\ E. x x e. A ) -> |^| A C_ A )

Distinct variable group:   x, A

Proof of Theorem trintssm

Step	Hyp	Ref	Expression
1		intss1 3651	. . . 4 |- ( x e. A -> |^| A C_ x )
2		trss 3884	. . . . 5 |- ( Tr A -> ( x e. A -> x C_ A ) )
3	2	com12 30	. . . 4 |- ( x e. A -> ( Tr A -> x C_ A ) )
4		sstr2 3006	. . . 4 |- ( |^| A C_ x -> ( x C_ A -> |^| A C_ A ) )
5	1, 3, 4	sylsyld 57	. . 3 |- ( x e. A -> ( Tr A -> |^| A C_ A ) )
6	5	exlimiv 1529	. 2 |- ( E. x x e. A -> ( Tr A -> |^| A C_ A ) )
7	6	impcom 123	1 |- ( ( Tr A /\ E. x x e. A ) -> |^| A C_ A )

Syntax hints:   -> wi 4   /\ wa 102   E.wex 1421   e. wcel 1433   C_ wss 2973   |^|cint 3636   Tr wtr 3875

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602  df-int 3637  df-tr 3876

This theorem is referenced by: (None)