Description: No class has 2-cycle membership loops. Theorem 7X(b) of Enderton p. 206. (Contributed by NM, 16-Oct-1996) (Revised by Mario Carneiro, 25-Jun-2015)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | en2lp | |- -. ( A e. B /\ B e. A ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | zfregfr | |- _E Fr _V |
|
| 2 | efrn2lp | |- ( ( _E Fr _V /\ ( A e. _V /\ B e. _V ) ) -> -. ( A e. B /\ B e. A ) ) |
|
| 3 | 1 2 | mpan | |- ( ( A e. _V /\ B e. _V ) -> -. ( A e. B /\ B e. A ) ) |
| 4 | elex | |- ( A e. B -> A e. _V ) |
|
| 5 | elex | |- ( B e. A -> B e. _V ) |
|
| 6 | 4 5 | anim12i | |- ( ( A e. B /\ B e. A ) -> ( A e. _V /\ B e. _V ) ) |
| 7 | 6 | con3i | |- ( -. ( A e. _V /\ B e. _V ) -> -. ( A e. B /\ B e. A ) ) |
| 8 | 3 7 | pm2.61i | |- -. ( A e. B /\ B e. A ) |