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Mirrors > Home > MPE Home > Th. List > epne3 | Structured version Visualization version GIF version |
Description: A well-founded class contains no 3-cycle loops. (Contributed by NM, 19-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.) |
Ref | Expression |
---|---|
epne3 | ⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fr3nr 7488 | . 2 ⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵 E 𝐶 ∧ 𝐶 E 𝐷 ∧ 𝐷 E 𝐵)) | |
2 | epelg 5460 | . . . . 5 ⊢ (𝐶 ∈ 𝐴 → (𝐵 E 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
3 | 2 | 3ad2ant2 1130 | . . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝐵 E 𝐶 ↔ 𝐵 ∈ 𝐶)) |
4 | epelg 5460 | . . . . 5 ⊢ (𝐷 ∈ 𝐴 → (𝐶 E 𝐷 ↔ 𝐶 ∈ 𝐷)) | |
5 | 4 | 3ad2ant3 1131 | . . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝐶 E 𝐷 ↔ 𝐶 ∈ 𝐷)) |
6 | epelg 5460 | . . . . 5 ⊢ (𝐵 ∈ 𝐴 → (𝐷 E 𝐵 ↔ 𝐷 ∈ 𝐵)) | |
7 | 6 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝐷 E 𝐵 ↔ 𝐷 ∈ 𝐵)) |
8 | 3, 5, 7 | 3anbi123d 1432 | . . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → ((𝐵 E 𝐶 ∧ 𝐶 E 𝐷 ∧ 𝐷 E 𝐵) ↔ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐵))) |
9 | 8 | adantl 484 | . 2 ⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐵 E 𝐶 ∧ 𝐶 E 𝐷 ∧ 𝐷 E 𝐵) ↔ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐵))) |
10 | 1, 9 | mtbid 326 | 1 ⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 class class class wbr 5058 E cep 5458 Fr wfr 5505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-eprel 5459 df-fr 5508 |
This theorem is referenced by: (None) |
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