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Theorem 2dom 6949
Description: A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.)
Assertion
Ref Expression
2dom  |-  ( 2o  ~<_  A  ->  E. x  e.  A  E. y  e.  A  -.  x  =  y )
Distinct variable group:    x, y, A

Proof of Theorem 2dom
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 df2o2 6509 . . . 4  |-  2o  =  { (/) ,  { (/) } }
21breq1i 4046 . . 3  |-  ( 2o  ~<_  A  <->  { (/) ,  { (/) } }  ~<_  A )
3 brdomi 6889 . . 3  |-  ( {
(/) ,  { (/) } }  ~<_  A  ->  E. f  f : { (/) ,  { (/) } } -1-1-> A )
42, 3sylbi 187 . 2  |-  ( 2o  ~<_  A  ->  E. f 
f : { (/) ,  { (/) } } -1-1-> A
)
5 f1f 5453 . . . . 5  |-  ( f : { (/) ,  { (/)
} } -1-1-> A  -> 
f : { (/) ,  { (/) } } --> A )
6 0ex 4166 . . . . . 6  |-  (/)  e.  _V
76prid1 3747 . . . . 5  |-  (/)  e.  { (/)
,  { (/) } }
8 ffvelrn 5679 . . . . 5  |-  ( ( f : { (/) ,  { (/) } } --> A  /\  (/) 
e.  { (/) ,  { (/)
} } )  -> 
( f `  (/) )  e.  A )
95, 7, 8sylancl 643 . . . 4  |-  ( f : { (/) ,  { (/)
} } -1-1-> A  -> 
( f `  (/) )  e.  A )
10 p0ex 4213 . . . . . 6  |-  { (/) }  e.  _V
1110prid2 3748 . . . . 5  |-  { (/) }  e.  { (/) ,  { (/)
} }
12 ffvelrn 5679 . . . . 5  |-  ( ( f : { (/) ,  { (/) } } --> A  /\  {
(/) }  e.  { (/) ,  { (/) } } )  ->  ( f `  { (/) } )  e.  A )
135, 11, 12sylancl 643 . . . 4  |-  ( f : { (/) ,  { (/)
} } -1-1-> A  -> 
( f `  { (/)
} )  e.  A
)
14 0nep0 4197 . . . . . 6  |-  (/)  =/=  { (/)
}
15 df-ne 2461 . . . . . 6  |-  ( (/)  =/=  { (/) }  <->  -.  (/)  =  { (/)
} )
1614, 15mpbi 199 . . . . 5  |-  -.  (/)  =  { (/)
}
17 f1fveq 5802 . . . . . 6  |-  ( ( f : { (/) ,  { (/) } } -1-1-> A  /\  ( (/)  e.  { (/) ,  { (/) } }  /\  {
(/) }  e.  { (/) ,  { (/) } } ) )  ->  ( (
f `  (/) )  =  ( f `  { (/)
} )  <->  (/)  =  { (/)
} ) )
187, 11, 17mpanr12 666 . . . . 5  |-  ( f : { (/) ,  { (/)
} } -1-1-> A  -> 
( ( f `  (/) )  =  ( f `
 { (/) } )  <->  (/)  =  { (/) } ) )
1916, 18mtbiri 294 . . . 4  |-  ( f : { (/) ,  { (/)
} } -1-1-> A  ->  -.  ( f `  (/) )  =  ( f `  { (/)
} ) )
20 eqeq1 2302 . . . . . 6  |-  ( x  =  ( f `  (/) )  ->  ( x  =  y  <->  ( f `  (/) )  =  y ) )
2120notbid 285 . . . . 5  |-  ( x  =  ( f `  (/) )  ->  ( -.  x  =  y  <->  -.  (
f `  (/) )  =  y ) )
22 eqeq2 2305 . . . . . 6  |-  ( y  =  ( f `  { (/) } )  -> 
( ( f `  (/) )  =  y  <->  ( f `  (/) )  =  ( f `  { (/) } ) ) )
2322notbid 285 . . . . 5  |-  ( y  =  ( f `  { (/) } )  -> 
( -.  ( f `
 (/) )  =  y  <->  -.  ( f `  (/) )  =  ( f `  { (/)
} ) ) )
2421, 23rspc2ev 2905 . . . 4  |-  ( ( ( f `  (/) )  e.  A  /\  ( f `
 { (/) } )  e.  A  /\  -.  ( f `  (/) )  =  ( f `  { (/)
} ) )  ->  E. x  e.  A  E. y  e.  A  -.  x  =  y
)
259, 13, 19, 24syl3anc 1182 . . 3  |-  ( f : { (/) ,  { (/)
} } -1-1-> A  ->  E. x  e.  A  E. y  e.  A  -.  x  =  y
)
2625exlimiv 1624 . 2  |-  ( E. f  f : { (/)
,  { (/) } } -1-1->
A  ->  E. x  e.  A  E. y  e.  A  -.  x  =  y )
274, 26syl 15 1  |-  ( 2o  ~<_  A  ->  E. x  e.  A  E. y  e.  A  -.  x  =  y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   (/)c0 3468   {csn 3653   {cpr 3654   class class class wbr 4039   -->wf 5267   -1-1->wf1 5268   ` cfv 5271   2oc2o 6489    ~<_ cdom 6877
This theorem is referenced by:  1sdom  7081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fv 5279  df-1o 6495  df-2o 6496  df-dom 6881
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