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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 9549 | . . 3 ⊢ E Fr V | |
2 | efrn2lp 5619 | . . 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 3465 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
5 | elex 3465 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V) | |
6 | 4, 5 | anim12i 614 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | 6 | con3i 154 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) |
8 | 3, 7 | pm2.61i 182 | 1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 ∈ wcel 2107 Vcvv 3447 E cep 5540 Fr wfr 5589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-reg 9536 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-eprel 5541 df-fr 5592 |
This theorem is referenced by: elnanel 9551 cnvepnep 9552 elnotel 9554 preleqALT 9561 suc11reg 9563 axunndlem1 10539 axacndlem5 10555 bj-nsnid 35591 tratrb 42910 tratrbVD 43235 |
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