MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  domen2 Structured version   Visualization version   Unicode version

Theorem domen2 7733
Description: Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
Assertion
Ref Expression
domen2  |-  ( A 
~~  B  ->  ( C  ~<_  A  <->  C  ~<_  B ) )

Proof of Theorem domen2
StepHypRef Expression
1 domentr 7646 . . 3  |-  ( ( C  ~<_  A  /\  A  ~~  B )  ->  C  ~<_  B )
21ancoms 460 . 2  |-  ( ( A  ~~  B  /\  C  ~<_  A )  ->  C  ~<_  B )
3 ensym 7636 . . 3  |-  ( A 
~~  B  ->  B  ~~  A )
4 domentr 7646 . . . 4  |-  ( ( C  ~<_  B  /\  B  ~~  A )  ->  C  ~<_  A )
54ancoms 460 . . 3  |-  ( ( B  ~~  A  /\  C  ~<_  B )  ->  C  ~<_  A )
63, 5sylan 479 . 2  |-  ( ( A  ~~  B  /\  C  ~<_  B )  ->  C  ~<_  A )
72, 6impbida 850 1  |-  ( A 
~~  B  ->  ( C  ~<_  A  <->  C  ~<_  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   class class class wbr 4395    ~~ cen 7584    ~<_ cdom 7585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-er 7381  df-en 7588  df-dom 7589
This theorem is referenced by:  infdiffi  8181  carddomi2  8422  numdom  8487  cdadom2  8635  infdif  8657  fin45  8840  fin67  8843  aleph1  9014  gchdomtri  9072  gchpwdom  9113  gchhar  9122  ctbnfien  35732
  Copyright terms: Public domain W3C validator