Theorem brdom2 4394

Description: Dominance in terms of strict dominance and equinumerosity. Theorem 22(iv) of [Suppes] p. 97.

Assertion

Ref	Expression
brdom2	|- (A ~<_ B <-> (A ~< B \/ A ~~ B))

Proof of Theorem brdom2

Step	Hyp	Ref	Expression
1		dfdom2 4390	. . 3 |- ~<_ = ( ~< u. ~~ )
2	1	eleq2i 1541	. 2 |- (<.A, B>. e. ~<_ <-> <.A, B>. e. ( ~< u. ~~ ))
3		df-br 2625	. 2 |- (A ~<_ B <-> <.A, B>. e. ~<_ )
4		df-br 2625	. . . 4 |- (A ~< B <-> <.A, B>. e. ~< )
5		df-br 2625	. . . 4 |- (A ~~ B <-> <.A, B>. e. ~~ )
6	4, 5	orbi12i 257	. . 3 |- ((A ~< B \/ A ~~ B) <-> (<.A, B>. e. ~< \/ <.A, B>. e. ~~ ))
7		elun 2176	. . 3 |- (<.A, B>. e. ( ~< u. ~~ ) <-> (<.A, B>. e. ~< \/ <.A, B>. e. ~~ ))
8	6, 7	bitr4 176	. 2 |- ((A ~< B \/ A ~~ B) <-> <.A, B>. e. ( ~< u. ~~ ))
9	2, 3, 8	3bitr4 183	1 |- (A ~<_ B <-> (A ~< B \/ A ~~ B))

Syntax hints:   <-> wb 146   \/ wo 222   e. wcel 960   u. cun 2048  <.cop 2415   class class class wbr 2624   ~~ cen 4370   ~<_ cdom 4371   ~< csdm 4372

This theorem is referenced by:  bren2 4395  domnsym 4469  sdomdomtr 4475  domsdomtr 4482  carddom 4846  entri 4849  entri2 4850

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-f1o 3203  df-en 4374  df-dom 4375  df-sdom 4376

Copyright terms: Public domain