![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 9052 | . . 3 ⊢ E Fr V | |
2 | efrn2lp 5501 | . . 3 ⊢ (( E Fr V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) | |
3 | 1, 2 | mpan 689 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) |
4 | elex 3459 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
5 | elex 3459 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V) | |
6 | 4, 5 | anim12i 615 | . . 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 185 | 1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 ∈ wcel 2111 Vcvv 3441 E cep 5429 Fr wfr 5475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-reg 9040 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-eprel 5430 df-fr 5478 |
This theorem is referenced by: elnanel 9054 cnvepnep 9055 elnotel 9057 preleqALT 9064 suc11reg 9066 axunndlem1 10006 axacndlem5 10022 bj-nsnid 34486 tratrb 41242 tratrbVD 41567 |
Copyright terms: Public domain | W3C validator |