Theorem efrn2lp 5604

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 5600	. 2 ⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵 E 𝐶 ∧ 𝐶 E 𝐵))
2		epelg 5524	. . . 4 ⊢ (𝐶 ∈ 𝐴 → (𝐵 E 𝐶 ↔ 𝐵 ∈ 𝐶))
3		epelg 5524	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (𝐶 E 𝐵 ↔ 𝐶 ∈ 𝐵))
4	2, 3	bi2anan9r 640	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐵 E 𝐶 ∧ 𝐶 E 𝐵) ↔ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐵)))
5	4	adantl 481	. 2 ⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝐵 E 𝐶 ∧ 𝐶 E 𝐵) ↔ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐵)))
6	1, 5	mtbid 324	1 ⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114   class class class wbr 5097   E cep 5522   Fr wfr 5573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-eprel 5523  df-fr 5576

This theorem is referenced by:  en2lp  9517