Theorem epne3 7768

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 7767	. 2 ⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵 E 𝐶 ∧ 𝐶 E 𝐷 ∧ 𝐷 E 𝐵))
2		epelg 5562	. . . . 5 ⊢ (𝐶 ∈ 𝐴 → (𝐵 E 𝐶 ↔ 𝐵 ∈ 𝐶))
3	2	3ad2ant2 1152	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝐵 E 𝐶 ↔ 𝐵 ∈ 𝐶))
4		epelg 5562	. . . . 5 ⊢ (𝐷 ∈ 𝐴 → (𝐶 E 𝐷 ↔ 𝐶 ∈ 𝐷))
5	4	3ad2ant3 1153	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝐶 E 𝐷 ↔ 𝐶 ∈ 𝐷))
6		epelg 5562	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → (𝐷 E 𝐵 ↔ 𝐷 ∈ 𝐵))
7	6	3ad2ant1 1151	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝐷 E 𝐵 ↔ 𝐷 ∈ 𝐵))
8	3, 5, 7	3anbi123d 1464	. . 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 486	. 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 327	1 ⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   ∈ wcel 2143   class class class wbr 5109   E cep 5560   Fr wfr 5611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-eprel 5561  df-fr 5614

This theorem is referenced by: (None)