Theorem trintssm 4147

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 3889	. . . 4 |- ( x e. A -> |^| A C_ x )
2		trss 4140	. . . . 5 |- ( Tr A -> ( x e. A -> x C_ A ) )
3	2	com12 30	. . . 4 |- ( x e. A -> ( Tr A -> x C_ A ) )
4		sstr2 3190	. . . 4 |- ( |^| A C_ x -> ( x C_ A -> |^| A C_ A ) )
5	1, 3, 4	sylsyld 58	. . 3 |- ( x e. A -> ( Tr A -> |^| A C_ A ) )
6	5	exlimiv 1612	. 2 |- ( E. x x e. A -> ( Tr A -> |^| A C_ A ) )
7	6	impcom 125	1 |- ( ( Tr A /\ E. x x e. A ) -> |^| A C_ A )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1506   e. wcel 2167   C_ wss 3157   |^|cint 3874   Tr wtr 4131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-ss 3170  df-uni 3840  df-int 3875  df-tr 4132

This theorem is referenced by: (None)