Theorem brdom2 7985

Description: Dominance in terms of strict dominance and equinumerosity. Theorem 22(iv) of [Suppes] p. 97. (Contributed by NM, 17-Jun-1998.)

Assertion

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

Proof of Theorem brdom2

Step	Hyp	Ref	Expression
1		dfdom2 7981	. . 3 |- ~<_ = ( ~< u. ~~ )
2	1	eleq2i 2693	. 2 |- ( <. A , B >. e. ~<_ <-> <. A , B >. e. ( ~< u. ~~ ) )
3		df-br 4654	. 2 |- ( A ~<_ B <-> <. A , B >. e. ~<_ )
4		df-br 4654	. . . 4 |- ( A ~< B <-> <. A , B >. e. ~< )
5		df-br 4654	. . . 4 |- ( A ~~ B <-> <. A , B >. e. ~~ )
6	4, 5	orbi12i 543	. . 3 |- ( ( A ~< B \/ A ~~ B ) <-> ( <. A , B >. e. ~< \/ <. A , B >. e. ~~ ) )
7		elun 3753	. . 3 |- ( <. A , B >. e. ( ~< u. ~~ ) <-> ( <. A , B >. e. ~< \/ <. A , B >. e. ~~ ) )
8	6, 7	bitr4i 267	. 2 |- ( ( A ~< B \/ A ~~ B ) <-> <. A , B >. e. ( ~< u. ~~ ) )
9	2, 3, 8	3bitr4i 292	1 |- ( A ~<_ B <-> ( A ~< B \/ A ~~ B ) )

Syntax hints:   <-> wb 196   \/ wo 383   e. wcel 1990   u. cun 3572   <.cop 4183   class class class wbr 4653   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954

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-ral 2917  df-rab 2921  df-v 3202  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-xp 5120  df-rel 5121  df-f1o 5895  df-en 7956  df-dom 7957  df-sdom 7958

This theorem is referenced by:  bren2  7986  domnsym  8086  modom  8161  carddom2  8803  axcc4dom  9263  entric  9379  entri2  9380  gchor  9449  frgpcyg  19922  iunmbl2  23325  dyadmbl  23368  padct  29497  volmeas  30294  ovoliunnfl  33451  ctbnfien  37382