Theorem trintssm 4203

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 3943	. . . 4 |- ( x e. A -> |^| A C_ x )
2		trss 4196	. . . . 5 |- ( Tr A -> ( x e. A -> x C_ A ) )
3	2	com12 30	. . . 4 |- ( x e. A -> ( Tr A -> x C_ A ) )
4		sstr2 3234	. . . 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 1646	. 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 1540   e. wcel 2202   C_ wss 3200   |^|cint 3928   Tr wtr 4187

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-in 3206  df-ss 3213  df-uni 3894  df-int 3929  df-tr 4188

This theorem is referenced by: (None)