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Theorem domen 6761
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
StepHypRef Expression
1 bren.1 . . 3  |-  B  e. 
_V
21brdom 6760 . 2  |-  ( A  ~<_  B  <->  E. f  f : A -1-1-> B )
3 vex 2730 . . . . . 6  |-  f  e. 
_V
43f11o 5363 . . . . 5  |-  ( f : A -1-1-> B  <->  E. x
( f : A -1-1-onto-> x  /\  x  C_  B ) )
54exbii 1580 . . . 4  |-  ( E. f  f : A -1-1-> B  <->  E. f E. x ( f : A -1-1-onto-> x  /\  x  C_  B ) )
6 excom 1765 . . . 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 242 . . 3  |-  ( E. f  f : A -1-1-> B  <->  E. x E. f ( f : A -1-1-onto-> x  /\  x  C_  B ) )
8 bren 6757 . . . . . 6  |-  ( A 
~~  x  <->  E. f 
f : A -1-1-onto-> x )
98anbi1i 679 . . . . 5  |-  ( ( A  ~~  x  /\  x  C_  B )  <->  ( E. f  f : A -1-1-onto-> x  /\  x  C_  B ) )
10 19.41v 2034 . . . . 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 245 . . . 4  |-  ( ( A  ~~  x  /\  x  C_  B )  <->  E. f
( f : A -1-1-onto-> x  /\  x  C_  B ) )
1211exbii 1580 . . 3  |-  ( E. x ( A  ~~  x  /\  x  C_  B
)  <->  E. x E. f
( f : A -1-1-onto-> x  /\  x  C_  B ) )
137, 12bitr4i 245 . 2  |-  ( E. f  f : A -1-1-> B  <->  E. x ( A  ~~  x  /\  x  C_  B
) )
142, 13bitri 242 1  |-  ( A  ~<_  B  <->  E. x ( A 
~~  x  /\  x  C_  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1537    e. wcel 1621   _Vcvv 2727    C_ wss 3078   class class class wbr 3920   -1-1->wf1 4589   -1-1-onto->wf1o 4591    ~~ cen 6746    ~<_ cdom 6747
This theorem is referenced by:  domeng  6762  infcntss  7015  cdainf  7702  ramub2  12935  ram0  12943
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-xp 4594  df-rel 4595  df-cnv 4596  df-dm 4598  df-rn 4599  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-en 6750  df-dom 6751
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