Theorem efrunt 31590

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 -> A. x e. A -. x e. x )

Distinct variable group:   x, A

Proof of Theorem efrunt

Step	Hyp	Ref	Expression
1		frirr 5091	. . 3 |- ( ( _E Fr A /\ x e. A ) -> -. x _E x )
2		epel 5032	. . 3 |- ( x _E x <-> x e. x )
3	1, 2	sylnib 318	. 2 |- ( ( _E Fr A /\ x e. A ) -> -. x e. x )
4	3	ralrimiva 2966	1 |- ( _E Fr A -> A. x e. A -. x e. x )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   class class class wbr 4653   _E cep 5028   Fr wfr 5070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-eprel 5029  df-fr 5073

This theorem is referenced by: (None)