Theorem dfdom2 3288

Description: Alternate definition of dominance.

Assertion

Ref	Expression
dfdom2	|- ~<_ = ( ~< u. ~~ )

Proof of Theorem dfdom2

Step	Hyp	Ref	Expression
1		df-sdom 3276	. . 3 |- ~< = ( ~<_ \ ~~ )
2	1	uneq2i 1608	. 2 |- ( ~~ u. ~< ) = ( ~~ u. ( ~<_ \ ~~ ))
3		uncom 1604	. 2 |- ( ~~ u. ~< ) = ( ~< u. ~~ )
4		enssdom 3287	. . 3 |- ~~ (_ ~<_
5		ssundif 1764	. . 3 |- ( ~~ (_ ~<_ <-> ( ~~ u. ( ~<_ \ ~~ )) = ~<_ )
6	4, 5	mpbi 164	. 2 |- ( ~~ u. ( ~<_ \ ~~ )) = ~<_
7	2, 3, 6	3eqtr3r 1125	1 |- ~<_ = ( ~< u. ~~ )

Syntax hints:   = wceq 1091   \ cdif 1484   u. cun 1485   (_ wss 1487   ~~ cen 3271   ~<_ cdom 3272   ~< csdm 3273

This theorem is referenced by:  brdom2 3292

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424  df-rel 2425  df-f1o 2437  df-en 3274  df-dom 3275  df-sdom 3276

metamath.org