Theorem wesn 5190

Description: Well-ordering of a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)

Assertion

Ref	Expression
wesn	|- ( Rel R -> ( R We { A } <-> -. A R A ) )

Proof of Theorem wesn

Step	Hyp	Ref	Expression
1		frsn 5189	. . 3 |- ( Rel R -> ( R Fr { A } <-> -. A R A ) )
2		sosn 5188	. . 3 |- ( Rel R -> ( R Or { A } <-> -. A R A ) )
3	1, 2	anbi12d 747	. 2 |- ( Rel R -> ( ( R Fr { A } /\ R Or { A } ) <-> ( -. A R A /\ -. A R A ) ) )
4		df-we 5075	. 2 |- ( R We { A } <-> ( R Fr { A } /\ R Or { A } ) )
5		pm4.24 675	. 2 |- ( -. A R A <-> ( -. A R A /\ -. A R A ) )
6	3, 4, 5	3bitr4g 303	1 |- ( Rel R -> ( R We { A } <-> -. A R A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   {csn 4177   class class class wbr 4653   Or wor 5034   Fr wfr 5070   We wwe 5072   Rel wrel 5119

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-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  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-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121

This theorem is referenced by:  0we1  7586  canthwe  9473