Theorem trintssmOLD 3959

Description: Obsolete version of trintssm 3958 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 2623	. . . 4 |- y e. _V
2	1	elint2 3701	. . 3 |- ( y e. |^| A <-> A. x e. A y e. x )
3		r19.2m 3373	. . . . 5 |- ( ( E. x x e. A /\ A. x e. A y e. x ) -> E. x e. A y e. x )
4	3	ex 114	. . . 4 |- ( E. x x e. A -> ( A. x e. A y e. x -> E. x e. A y e. x ) )
5		trel 3949	. . . . . 6 |- ( Tr A -> ( ( y e. x /\ x e. A ) -> y e. A ) )
6	5	expcomd 1376	. . . . 5 |- ( Tr A -> ( x e. A -> ( y e. x -> y e. A ) ) )
7	6	rexlimdv 2489	. . . 4 |- ( Tr A -> ( E. x e. A y e. x -> y e. A ) )
8	4, 7	sylan9 402	. . 3 |- ( ( E. x x e. A /\ Tr A ) -> ( A. x e. A y e. x -> y e. A ) )
9	2, 8	syl5bi 151	. 2 |- ( ( E. x x e. A /\ Tr A ) -> ( y e. |^| A -> y e. A ) )
10	9	ssrdv 3032	1 |- ( ( E. x x e. A /\ Tr A ) -> |^| A C_ A )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1427   e. wcel 1439   A.wral 2360   E.wrex 2361   C_ wss 3000   |^|cint 3694   Tr wtr 3942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-in 3006  df-ss 3013  df-uni 3660  df-int 3695  df-tr 3943

This theorem is referenced by: (None)