Theorem dfdom2 4384

Description: Alternate definition of dominance.

Assertion

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

Proof of Theorem dfdom2

Step	Hyp	Ref	Expression
1		df-sdom 4370	. . 3 |- ~< = ( ~<_ \ ~~ )
2	1	uneq2i 2181	. 2 |- ( ~~ u. ~< ) = ( ~~ u. ( ~<_ \ ~~ ))
3		uncom 2176	. 2 |- ( ~~ u. ~< ) = ( ~< u. ~~ )
4		enssdom 4383	. . 3 |- ~~ (_ ~<_
5		undif 2343	. . 3 |- ( ~~ (_ ~<_ <-> ( ~~ u. ( ~<_ \ ~~ )) = ~<_ )
6	4, 5	mpbi 189	. 2 |- ( ~~ u. ( ~<_ \ ~~ )) = ~<_
7	2, 3, 6	3eqtr3r 1504	1 |- ~<_ = ( ~< u. ~~ )

Syntax hints:   = wceq 956   \ cdif 2044   u. cun 2045   (_ wss 2047   ~~ cen 4364   ~<_ cdom 4365   ~< csdm 4366

This theorem is referenced by:  brdom2 4388

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-opab 2667  df-xp 3184  df-rel 3185  df-f1o 3197  df-en 4368  df-dom 4369  df-sdom 4370

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