Theorem trintssmOLD 3892

Description: Obsolete version of trintssm 3891 as of 30-Oct-2021. (Contributed by Jim Kingdon, 22-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   x, A

Proof of Theorem trintssmOLD

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2604	. . . 4 |- y e. _V
2	1	elint2 3643	. . 3 |- ( y e. |^| A <-> A. x e. A y e. x )
3		r19.2m 3329	. . . . 5 |- ( ( E. x x e. A /\ A. x e. A y e. x ) -> E. x e. A y e. x )
4	3	ex 113	. . . 4 |- ( E. x x e. A -> ( A. x e. A y e. x -> E. x e. A y e. x ) )
5		trel 3882	. . . . . 6 |- ( Tr A -> ( ( y e. x /\ x e. A ) -> y e. A ) )
6	5	expcomd 1370	. . . . 5 |- ( Tr A -> ( x e. A -> ( y e. x -> y e. A ) ) )
7	6	rexlimdv 2476	. . . 4 |- ( Tr A -> ( E. x e. A y e. x -> y e. A ) )
8	4, 7	sylan9 401	. . 3 |- ( ( E. x x e. A /\ Tr A ) -> ( A. x e. A y e. x -> y e. A ) )
9	2, 8	syl5bi 150	. 2 |- ( ( E. x x e. A /\ Tr A ) -> ( y e. |^| A -> y e. A ) )
10	9	ssrdv 3005	1 |- ( ( E. x x e. A /\ Tr A ) -> |^| A C_ A )

Syntax hints:   -> wi 4   /\ wa 102   E.wex 1421   e. wcel 1433   A.wral 2348   E.wrex 2349   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-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602  df-int 3637  df-tr 3876

This theorem is referenced by: (None)