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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 8785 | . . 3 ⊢ E Fr V | |
2 | efrn2lp 5328 | . . 3 ⊢ (( E Fr V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) | |
3 | 1, 2 | mpan 681 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) |
4 | elex 3429 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
5 | elex 3429 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V) | |
6 | 4, 5 | anim12i 606 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | 6 | con3i 152 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) |
8 | 3, 7 | pm2.61i 177 | 1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 386 ∈ wcel 2164 Vcvv 3414 E cep 5256 Fr wfr 5302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 ax-reg 8773 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-eprel 5257 df-fr 5305 |
This theorem is referenced by: cnvepnep 8787 elnotel 8789 preleqALT 8796 preleqOLD 8798 suc11reg 8800 axunndlem1 9739 axacndlem5 9755 bj-nsnid 33630 tratrb 39579 tratrbVD 39914 |
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