Theorem epne3 7715

Description: A well-founded class contains no 3-cycle loops. (Contributed by NM, 19-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)

Assertion

Ref	Expression
epne3	⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐵))

Proof of Theorem epne3

Step	Hyp	Ref	Expression
1		fr3nr 7714	. 2 ⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵 E 𝐶 ∧ 𝐶 E 𝐷 ∧ 𝐷 E 𝐵))
2		epelg 5522	. . . . 5 ⊢ (𝐶 ∈ 𝐴 → (𝐵 E 𝐶 ↔ 𝐵 ∈ 𝐶))
3	2	3ad2ant2 1134	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝐵 E 𝐶 ↔ 𝐵 ∈ 𝐶))
4		epelg 5522	. . . . 5 ⊢ (𝐷 ∈ 𝐴 → (𝐶 E 𝐷 ↔ 𝐶 ∈ 𝐷))
5	4	3ad2ant3 1135	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝐶 E 𝐷 ↔ 𝐶 ∈ 𝐷))
6		epelg 5522	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → (𝐷 E 𝐵 ↔ 𝐷 ∈ 𝐵))
7	6	3ad2ant1 1133	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝐷 E 𝐵 ↔ 𝐷 ∈ 𝐵))
8	3, 5, 7	3anbi123d 1438	. . 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 481	. 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 324	1 ⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2113   class class class wbr 5095   E cep 5520   Fr wfr 5571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-eprel 5521  df-fr 5574

This theorem is referenced by: (None)