Theorem efrn2lp 5646

Description: A well-founded class contains no 2-cycle loops. (Contributed by NM, 19-Apr-1994.)

Assertion

Ref	Expression
efrn2lp	⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐵))

Proof of Theorem efrn2lp

Step	Hyp	Ref	Expression
1		fr2nr 5642	. 2 ⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵 E 𝐶 ∧ 𝐶 E 𝐵))
2		epelg 5566	. . . 4 ⊢ (𝐶 ∈ 𝐴 → (𝐵 E 𝐶 ↔ 𝐵 ∈ 𝐶))
3		epelg 5566	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (𝐶 E 𝐵 ↔ 𝐶 ∈ 𝐵))
4	2, 3	bi2anan9r 650	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐵 E 𝐶 ∧ 𝐶 E 𝐵) ↔ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐵)))
5	4	adantl 486	. 2 ⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝐵 E 𝐶 ∧ 𝐶 E 𝐵) ↔ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐵)))
6	1, 5	mtbid 327	1 ⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2150   class class class wbr 5114   E cep 5564   Fr wfr 5615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-eprel 5565  df-fr 5618

This theorem is referenced by:  en2lp  9578