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Theorem domen 7112
Description: Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146. (Contributed by NM, 15-Jun-1998.)
Hypothesis
Ref Expression
bren.1  |-  B  e. 
_V
Assertion
Ref Expression
domen  |-  ( A  ~<_  B  <->  E. x ( A 
~~  x  /\  x  C_  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem domen
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 bren.1 . . 3  |-  B  e. 
_V
21brdom 7111 . 2  |-  ( A  ~<_  B  <->  E. f  f : A -1-1-> B )
3 vex 2951 . . . . . 6  |-  f  e. 
_V
43f11o 5699 . . . . 5  |-  ( f : A -1-1-> B  <->  E. x
( f : A -1-1-onto-> x  /\  x  C_  B ) )
54exbii 1592 . . . 4  |-  ( E. f  f : A -1-1-> B  <->  E. f E. x ( f : A -1-1-onto-> x  /\  x  C_  B ) )
6 excom 1756 . . . 4  |-  ( E. f E. x ( f : A -1-1-onto-> x  /\  x  C_  B )  <->  E. x E. f ( f : A -1-1-onto-> x  /\  x  C_  B ) )
75, 6bitri 241 . . 3  |-  ( E. f  f : A -1-1-> B  <->  E. x E. f ( f : A -1-1-onto-> x  /\  x  C_  B ) )
8 bren 7108 . . . . . 6  |-  ( A 
~~  x  <->  E. f 
f : A -1-1-onto-> x )
98anbi1i 677 . . . . 5  |-  ( ( A  ~~  x  /\  x  C_  B )  <->  ( E. f  f : A -1-1-onto-> x  /\  x  C_  B ) )
10 19.41v 1924 . . . . 5  |-  ( E. f ( f : A -1-1-onto-> x  /\  x  C_  B )  <->  ( E. f  f : A -1-1-onto-> x  /\  x  C_  B ) )
119, 10bitr4i 244 . . . 4  |-  ( ( A  ~~  x  /\  x  C_  B )  <->  E. f
( f : A -1-1-onto-> x  /\  x  C_  B ) )
1211exbii 1592 . . 3  |-  ( E. x ( A  ~~  x  /\  x  C_  B
)  <->  E. x E. f
( f : A -1-1-onto-> x  /\  x  C_  B ) )
137, 12bitr4i 244 . 2  |-  ( E. f  f : A -1-1-> B  <->  E. x ( A  ~~  x  /\  x  C_  B
) )
142, 13bitri 241 1  |-  ( A  ~<_  B  <->  E. x ( A 
~~  x  /\  x  C_  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    e. wcel 1725   _Vcvv 2948    C_ wss 3312   class class class wbr 4204   -1-1->wf1 5442   -1-1-onto->wf1o 5444    ~~ cen 7097    ~<_ cdom 7098
This theorem is referenced by:  domeng  7113  infcntss  7371  cdainf  8061  ramub2  13370  ram0  13378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4875  df-rel 4876  df-cnv 4877  df-dm 4879  df-rn 4880  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-en 7101  df-dom 7102
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