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Mirrors > Home > ILE Home > Th. List > efrirr | GIF version |
Description: Irreflexivity of the epsilon relation: a class founded by epsilon is not a member of itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.) |
Ref | Expression |
---|---|
efrirr | ⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frirrg 4105 | . . . 4 ⊢ (( E Fr 𝐴 ∧ 𝐴 ∈ 𝐴 ∧ 𝐴 ∈ 𝐴) → ¬ 𝐴 E 𝐴) | |
2 | 1 | 3anidm23 1228 | . . 3 ⊢ (( E Fr 𝐴 ∧ 𝐴 ∈ 𝐴) → ¬ 𝐴 E 𝐴) |
3 | epelg 4045 | . . . 4 ⊢ (𝐴 ∈ 𝐴 → (𝐴 E 𝐴 ↔ 𝐴 ∈ 𝐴)) | |
4 | 3 | adantl 271 | . . 3 ⊢ (( E Fr 𝐴 ∧ 𝐴 ∈ 𝐴) → (𝐴 E 𝐴 ↔ 𝐴 ∈ 𝐴)) |
5 | 2, 4 | mtbid 629 | . 2 ⊢ (( E Fr 𝐴 ∧ 𝐴 ∈ 𝐴) → ¬ 𝐴 ∈ 𝐴) |
6 | 5 | pm2.01da 597 | 1 ⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∈ wcel 1433 class class class wbr 3785 E cep 4042 Fr wfr 4083 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-eprel 4044 df-frfor 4086 df-frind 4087 |
This theorem is referenced by: tz7.2 4109 |
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