Theorem efrunt 35883

Description: If 𝐴 is well-founded by E, then it is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)

Assertion

Ref	Expression
efrunt	⊢ ( E Fr 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥)

Distinct variable group:   𝑥,𝐴

Proof of Theorem efrunt

Step	Hyp	Ref	Expression
1		frirr 5596	. . 3 ⊢ (( E Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 E 𝑥)
2		epel 5523	. . 3 ⊢ (𝑥 E 𝑥 ↔ 𝑥 ∈ 𝑥)
3	1, 2	sylnib 328	. 2 ⊢ (( E Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 ∈ 𝑥)
4	3	ralrimiva 3127	1 ⊢ ( E Fr 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2114  ∀wral 3049   class class class wbr 5074   E cep 5519   Fr wfr 5570

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 5220  ax-pr 5364

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 2931  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-br 5075  df-opab 5137  df-eprel 5520  df-fr 5573

This theorem is referenced by: (None)