MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bren2 Structured version   Unicode version

Theorem bren2 7539
Description: Equinumerosity expressed in terms of dominance and strict dominance. (Contributed by NM, 23-Oct-2004.)
Assertion
Ref Expression
bren2  |-  ( A 
~~  B  <->  ( A  ~<_  B  /\  -.  A  ~<  B ) )

Proof of Theorem bren2
StepHypRef Expression
1 endom 7535 . . 3  |-  ( A 
~~  B  ->  A  ~<_  B )
2 sdomnen 7537 . . . 4  |-  ( A 
~<  B  ->  -.  A  ~~  B )
32con2i 120 . . 3  |-  ( A 
~~  B  ->  -.  A  ~<  B )
41, 3jca 530 . 2  |-  ( A 
~~  B  ->  ( A  ~<_  B  /\  -.  A  ~<  B ) )
5 brdom2 7538 . . . 4  |-  ( A  ~<_  B  <->  ( A  ~<  B  \/  A  ~~  B
) )
65biimpi 194 . . 3  |-  ( A  ~<_  B  ->  ( A  ~<  B  \/  A  ~~  B ) )
76orcanai 911 . 2  |-  ( ( A  ~<_  B  /\  -.  A  ~<  B )  ->  A  ~~  B )
84, 7impbii 188 1  |-  ( A 
~~  B  <->  ( A  ~<_  B  /\  -.  A  ~<  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 366    /\ wa 367   class class class wbr 4439    ~~ cen 7506    ~<_ cdom 7507    ~< csdm 7508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-f1o 5577  df-en 7510  df-dom 7511  df-sdom 7512
This theorem is referenced by:  marypha1lem  7885  tskwe  8322  infxpenlem  8382  cdainflem  8562  axcclem  8828  alephsuc3  8946  gchen1  8992  gchen2  8993  inatsk  9145  ufilen  20597  dirith2  23911  mblfinlem1  30291
  Copyright terms: Public domain W3C validator