Theorem trintssOLD 4770

Description: Obsolete version of trintss 4769 as of 30-Oct-2021. (Contributed by Scott Fenton, 3-Mar-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem trintssOLD

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . 4 |- y e. _V
2	1	elint2 4482	. . 3 |- ( y e. |^| A <-> A. x e. A y e. x )
3		r19.2z 4060	. . . . 5 |- ( ( A =/= (/) /\ A. x e. A y e. x ) -> E. x e. A y e. x )
4	3	ex 450	. . . 4 |- ( A =/= (/) -> ( A. x e. A y e. x -> E. x e. A y e. x ) )
5		trel 4759	. . . . . 6 |- ( Tr A -> ( ( y e. x /\ x e. A ) -> y e. A ) )
6	5	expcomd 454	. . . . 5 |- ( Tr A -> ( x e. A -> ( y e. x -> y e. A ) ) )
7	6	rexlimdv 3030	. . . 4 |- ( Tr A -> ( E. x e. A y e. x -> y e. A ) )
8	4, 7	sylan9 689	. . 3 |- ( ( A =/= (/) /\ Tr A ) -> ( A. x e. A y e. x -> y e. A ) )
9	2, 8	syl5bi 232	. 2 |- ( ( A =/= (/) /\ Tr A ) -> ( y e. |^| A -> y e. A ) )
10	9	ssrdv 3609	1 |- ( ( A =/= (/) /\ Tr A ) -> |^| A C_ A )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   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-rex 2918  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)