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Mirrors > Home > MPE Home > Th. List > en2lp | Structured version Visualization version GIF version |
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 |
---|---|
en2lp | ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zfregfr 9068 | . . 3 ⊢ E Fr V | |
2 | efrn2lp 5537 | . . 3 ⊢ (( E Fr V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) | |
3 | 1, 2 | mpan 688 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) |
4 | elex 3512 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
5 | elex 3512 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V) | |
6 | 4, 5 | anim12i 614 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | 6 | con3i 157 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) |
8 | 3, 7 | pm2.61i 184 | 1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 ∈ wcel 2114 Vcvv 3494 E cep 5464 Fr wfr 5511 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-reg 9056 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-eprel 5465 df-fr 5514 |
This theorem is referenced by: elnanel 9070 cnvepnep 9071 elnotel 9073 preleqALT 9080 suc11reg 9082 axunndlem1 10017 axacndlem5 10033 bj-nsnid 34365 tratrb 40890 tratrbVD 41215 |
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