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 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zfregfr | ⊢ E Fr V | |
2 | efrn2lp | ⊢ ( ( E Fr V ∧ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) → ¬ ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴 ) ) | |
3 | 1 2 | mpan | ⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ¬ ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴 ) ) |
4 | elex | ⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ V ) | |
5 | elex | ⊢ ( 𝐵 ∈ 𝐴 → 𝐵 ∈ V ) | |
6 | 4 5 | anim12i | ⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴 ) → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) |
7 | 6 | con3i | ⊢ ( ¬ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ¬ ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴 ) ) |
8 | 3 7 | pm2.61i | ⊢ ¬ ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴 ) |