Theorem trintss 4769

Description: Any nonempty transitive class includes its intersection. Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the nonemptiness hypothesis). (Contributed by Scott Fenton, 3-Mar-2011.) (Proof shortened by Andrew Salmon, 14-Nov-2011.)

Assertion

Ref	Expression
trintss	|- ( ( Tr A /\ A =/= (/) ) -> |^| A C_ A )

Proof of Theorem trintss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3931	. . 3 |- ( A =/= (/) <-> E. x x e. A )
2		intss1 4492	. . . . 5 |- ( x e. A -> |^| A C_ x )
3		trss 4761	. . . . . 6 |- ( Tr A -> ( x e. A -> x C_ A ) )
4	3	com12 32	. . . . 5 |- ( x e. A -> ( Tr A -> x C_ A ) )
5		sstr2 3610	. . . . 5 |- ( |^| A C_ x -> ( x C_ A -> |^| A C_ A ) )
6	2, 4, 5	sylsyld 61	. . . 4 |- ( x e. A -> ( Tr A -> |^| A C_ A ) )
7	6	exlimiv 1858	. . 3 |- ( E. x x e. A -> ( Tr A -> |^| A C_ A ) )
8	1, 7	sylbi 207	. 2 |- ( A =/= (/) -> ( Tr A -> |^| A C_ A ) )
9	8	impcom 446	1 |- ( ( Tr A /\ A =/= (/) ) -> |^| A C_ A )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   |^|cint 4475   Tr wtr 4752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-uni 4437  df-int 4476  df-tr 4753

This theorem is referenced by: (None)