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Theorem 2dom 7660
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 7214 . . . 4  |-  2o  =  { (/) ,  { (/) } }
21breq1i 4402 . . 3  |-  ( 2o  ~<_  A  <->  { (/) ,  { (/) } }  ~<_  A )
3 brdomi 7598 . . 3  |-  ( {
(/) ,  { (/) } }  ~<_  A  ->  E. f  f : { (/) ,  { (/) } } -1-1-> A )
42, 3sylbi 200 . 2  |-  ( 2o  ~<_  A  ->  E. f 
f : { (/) ,  { (/) } } -1-1-> A
)
5 f1f 5792 . . . . 5  |-  ( f : { (/) ,  { (/)
} } -1-1-> A  -> 
f : { (/) ,  { (/) } } --> A )
6 0ex 4528 . . . . . 6  |-  (/)  e.  _V
76prid1 4071 . . . . 5  |-  (/)  e.  { (/)
,  { (/) } }
8 ffvelrn 6035 . . . . 5  |-  ( ( f : { (/) ,  { (/) } } --> A  /\  (/) 
e.  { (/) ,  { (/)
} } )  -> 
( f `  (/) )  e.  A )
95, 7, 8sylancl 675 . . . 4  |-  ( f : { (/) ,  { (/)
} } -1-1-> A  -> 
( f `  (/) )  e.  A )
10 p0ex 4588 . . . . . 6  |-  { (/) }  e.  _V
1110prid2 4072 . . . . 5  |-  { (/) }  e.  { (/) ,  { (/)
} }
12 ffvelrn 6035 . . . . 5  |-  ( ( f : { (/) ,  { (/) } } --> A  /\  {
(/) }  e.  { (/) ,  { (/) } } )  ->  ( f `  { (/) } )  e.  A )
135, 11, 12sylancl 675 . . . 4  |-  ( f : { (/) ,  { (/)
} } -1-1-> A  -> 
( f `  { (/)
} )  e.  A
)
14 0nep0 4572 . . . . . 6  |-  (/)  =/=  { (/)
}
1514neii 2645 . . . . 5  |-  -.  (/)  =  { (/)
}
16 f1fveq 6181 . . . . . 6  |-  ( ( f : { (/) ,  { (/) } } -1-1-> A  /\  ( (/)  e.  { (/) ,  { (/) } }  /\  {
(/) }  e.  { (/) ,  { (/) } } ) )  ->  ( (
f `  (/) )  =  ( f `  { (/)
} )  <->  (/)  =  { (/)
} ) )
177, 11, 16mpanr12 699 . . . . 5  |-  ( f : { (/) ,  { (/)
} } -1-1-> A  -> 
( ( f `  (/) )  =  ( f `
 { (/) } )  <->  (/)  =  { (/) } ) )
1815, 17mtbiri 310 . . . 4  |-  ( f : { (/) ,  { (/)
} } -1-1-> A  ->  -.  ( f `  (/) )  =  ( f `  { (/)
} ) )
19 eqeq1 2475 . . . . . 6  |-  ( x  =  ( f `  (/) )  ->  ( x  =  y  <->  ( f `  (/) )  =  y ) )
2019notbid 301 . . . . 5  |-  ( x  =  ( f `  (/) )  ->  ( -.  x  =  y  <->  -.  (
f `  (/) )  =  y ) )
21 eqeq2 2482 . . . . . 6  |-  ( y  =  ( f `  { (/) } )  -> 
( ( f `  (/) )  =  y  <->  ( f `  (/) )  =  ( f `  { (/) } ) ) )
2221notbid 301 . . . . 5  |-  ( y  =  ( f `  { (/) } )  -> 
( -.  ( f `
 (/) )  =  y  <->  -.  ( f `  (/) )  =  ( f `  { (/)
} ) ) )
2320, 22rspc2ev 3149 . . . 4  |-  ( ( ( f `  (/) )  e.  A  /\  ( f `
 { (/) } )  e.  A  /\  -.  ( f `  (/) )  =  ( f `  { (/)
} ) )  ->  E. x  e.  A  E. y  e.  A  -.  x  =  y
)
249, 13, 18, 23syl3anc 1292 . . 3  |-  ( f : { (/) ,  { (/)
} } -1-1-> A  ->  E. x  e.  A  E. y  e.  A  -.  x  =  y
)
2524exlimiv 1784 . 2  |-  ( E. f  f : { (/)
,  { (/) } } -1-1->
A  ->  E. x  e.  A  E. y  e.  A  -.  x  =  y )
264, 25syl 17 1  |-  ( 2o  ~<_  A  ->  E. x  e.  A  E. y  e.  A  -.  x  =  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    = wceq 1452   E.wex 1671    e. wcel 1904   E.wrex 2757   (/)c0 3722   {csn 3959   {cpr 3961   class class class wbr 4395   -->wf 5585   -1-1->wf1 5586   ` cfv 5589   2oc2o 7194    ~<_ 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-sbc 3256  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-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fv 5597  df-1o 7200  df-2o 7201  df-dom 7589
This theorem is referenced by:  1sdom  7793
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