Theorem efrunt 35700

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 5614	. . 3 ⊢ (( E Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 E 𝑥)
2		epel 5541	. . 3 ⊢ (𝑥 E 𝑥 ↔ 𝑥 ∈ 𝑥)
3	1, 2	sylnib 328	. 2 ⊢ (( E Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 ∈ 𝑥)
4	3	ralrimiva 3125	1 ⊢ ( E Fr 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀wral 3044   class class class wbr 5107   E cep 5537   Fr wfr 5588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-eprel 5538  df-fr 5591

This theorem is referenced by: (None)