Theorem trintssm 4037

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 3781	. . . 4 |- ( x e. A -> |^| A C_ x )
2		trss 4030	. . . . 5 |- ( Tr A -> ( x e. A -> x C_ A ) )
3	2	com12 30	. . . 4 |- ( x e. A -> ( Tr A -> x C_ A ) )
4		sstr2 3099	. . . 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 1577	. 2 |- ( E. x x e. A -> ( Tr A -> |^| A C_ A ) )
7	6	impcom 124	1 |- ( ( Tr A /\ E. x x e. A ) -> |^| A C_ A )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1468   e. wcel 1480   C_ wss 3066   |^|cint 3766   Tr wtr 4021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-in 3072  df-ss 3079  df-uni 3732  df-int 3767  df-tr 4022

This theorem is referenced by: (None)