What is the smallest size emt conduit that can be used with three 14 gauge thwn wires and four 6 gauge thwn wires?

This is per the NEC Handbook, 2005 edition. Your municipality may use different rules. I am not an electrician.

Per Table 5, chapter 9:

Approximate area of 14 ga. THWN is 0.0097 sq. in.

Approximate area of 6 ga. THWN is 0.0507 sq. in.

0.0097 * 3 = 0.0291

0.0507 * 4 = 0.2028

...total area 0.2319 sq. in.

Per table 1 chapter 9, you are allowed 40% fill for more than two wires in EMT. So, 0.2319 / 0.4 = 5.7975 sq. in. minimum conduit area, which means you'd have to use 2-1/2 inch trade size EMT, which is 5.858 sq. in.

Hope this helps.