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Theorem dcomex 8073
Description: The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.)
Assertion
Ref Expression
dcomex  |-  om  e.  _V

Proof of Theorem dcomex
Dummy variables  t 
s  x  f  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 4216 . . 3  |-  { <. 1o ,  1o >. }  e.  _V
2 1on 6486 . . . . . . . . . 10  |-  1o  e.  On
32elexi 2797 . . . . . . . . 9  |-  1o  e.  _V
43, 3fvsn 5713 . . . . . . . 8  |-  ( {
<. 1o ,  1o >. } `
 1o )  =  1o
53, 3funsn 5300 . . . . . . . . 9  |-  Fun  { <. 1o ,  1o >. }
63snid 3667 . . . . . . . . . 10  |-  1o  e.  { 1o }
73dmsnop 5147 . . . . . . . . . 10  |-  dom  { <. 1o ,  1o >. }  =  { 1o }
86, 7eleqtrri 2356 . . . . . . . . 9  |-  1o  e.  dom  { <. 1o ,  1o >. }
9 funbrfvb 5565 . . . . . . . . 9  |-  ( ( Fun  { <. 1o ,  1o >. }  /\  1o  e.  dom  { <. 1o ,  1o >. } )  -> 
( ( { <. 1o ,  1o >. } `  1o )  =  1o  <->  1o { <. 1o ,  1o >. } 1o ) )
105, 8, 9mp2an 653 . . . . . . . 8  |-  ( ( { <. 1o ,  1o >. } `  1o )  =  1o  <->  1o { <. 1o ,  1o >. } 1o )
114, 10mpbi 199 . . . . . . 7  |-  1o { <. 1o ,  1o >. } 1o
12 breq12 4028 . . . . . . . 8  |-  ( ( s  =  1o  /\  t  =  1o )  ->  ( s { <. 1o ,  1o >. } t  <-> 
1o { <. 1o ,  1o >. } 1o ) )
133, 3, 12spc2ev 2876 . . . . . . 7  |-  ( 1o { <. 1o ,  1o >. } 1o  ->  E. s E. t  s { <. 1o ,  1o >. } t )
1411, 13ax-mp 8 . . . . . 6  |-  E. s E. t  s { <. 1o ,  1o >. } t
15 breq 4025 . . . . . . 7  |-  ( x  =  { <. 1o ,  1o >. }  ->  (
s x t  <->  s { <. 1o ,  1o >. } t ) )
16152exbidv 1614 . . . . . 6  |-  ( x  =  { <. 1o ,  1o >. }  ->  ( E. s E. t  s x t  <->  E. s E. t  s { <. 1o ,  1o >. } t ) )
1714, 16mpbiri 224 . . . . 5  |-  ( x  =  { <. 1o ,  1o >. }  ->  E. s E. t  s x
t )
18 ssid 3197 . . . . . . 7  |-  { 1o }  C_  { 1o }
193rnsnop 5153 . . . . . . 7  |-  ran  { <. 1o ,  1o >. }  =  { 1o }
2018, 19, 73sstr4i 3217 . . . . . 6  |-  ran  { <. 1o ,  1o >. } 
C_  dom  { <. 1o ,  1o >. }
21 rneq 4904 . . . . . . 7  |-  ( x  =  { <. 1o ,  1o >. }  ->  ran  x  =  ran  { <. 1o ,  1o >. } )
22 dmeq 4879 . . . . . . 7  |-  ( x  =  { <. 1o ,  1o >. }  ->  dom  x  =  dom  { <. 1o ,  1o >. } )
2321, 22sseq12d 3207 . . . . . 6  |-  ( x  =  { <. 1o ,  1o >. }  ->  ( ran  x  C_  dom  x  <->  ran  { <. 1o ,  1o >. }  C_  dom  { <. 1o ,  1o >. } ) )
2420, 23mpbiri 224 . . . . 5  |-  ( x  =  { <. 1o ,  1o >. }  ->  ran  x  C_  dom  x )
25 pm5.5 326 . . . . 5  |-  ( ( E. s E. t 
s x t  /\  ran  x  C_  dom  x )  ->  ( ( ( E. s E. t 
s x t  /\  ran  x  C_  dom  x )  ->  E. f A. n  e.  om  ( f `  n ) x ( f `  suc  n
) )  <->  E. f A. n  e.  om  ( f `  n
) x ( f `
 suc  n )
) )
2617, 24, 25syl2anc 642 . . . 4  |-  ( x  =  { <. 1o ,  1o >. }  ->  (
( ( E. s E. t  s x
t  /\  ran  x  C_  dom  x )  ->  E. f A. n  e.  om  ( f `  n
) x ( f `
 suc  n )
)  <->  E. f A. n  e.  om  ( f `  n ) x ( f `  suc  n
) ) )
27 breq 4025 . . . . . 6  |-  ( x  =  { <. 1o ,  1o >. }  ->  (
( f `  n
) x ( f `
 suc  n )  <->  ( f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n ) ) )
2827ralbidv 2563 . . . . 5  |-  ( x  =  { <. 1o ,  1o >. }  ->  ( A. n  e.  om  ( f `  n
) x ( f `
 suc  n )  <->  A. n  e.  om  (
f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n ) ) )
2928exbidv 1612 . . . 4  |-  ( x  =  { <. 1o ,  1o >. }  ->  ( E. f A. n  e. 
om  ( f `  n ) x ( f `  suc  n
)  <->  E. f A. n  e.  om  ( f `  n ) { <. 1o ,  1o >. }  (
f `  suc  n ) ) )
3026, 29bitrd 244 . . 3  |-  ( x  =  { <. 1o ,  1o >. }  ->  (
( ( E. s E. t  s x
t  /\  ran  x  C_  dom  x )  ->  E. f A. n  e.  om  ( f `  n
) x ( f `
 suc  n )
)  <->  E. f A. n  e.  om  ( f `  n ) { <. 1o ,  1o >. }  (
f `  suc  n ) ) )
31 ax-dc 8072 . . 3  |-  ( ( E. s E. t 
s x t  /\  ran  x  C_  dom  x )  ->  E. f A. n  e.  om  ( f `  n ) x ( f `  suc  n
) )
321, 30, 31vtocl 2838 . 2  |-  E. f A. n  e.  om  ( f `  n
) { <. 1o ,  1o >. }  ( f `
 suc  n )
33 1n0 6494 . . . . . . . 8  |-  1o  =/=  (/)
34 df-br 4024 . . . . . . . . 9  |-  ( ( f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n )  <->  <. ( f `
 n ) ,  ( f `  suc  n ) >.  e.  { <. 1o ,  1o >. } )
35 elsni 3664 . . . . . . . . . 10  |-  ( <.
( f `  n
) ,  ( f `
 suc  n ) >.  e.  { <. 1o ,  1o >. }  ->  <. (
f `  n ) ,  ( f `  suc  n ) >.  =  <. 1o ,  1o >. )
36 fvex 5539 . . . . . . . . . . 11  |-  ( f `
 n )  e. 
_V
37 fvex 5539 . . . . . . . . . . 11  |-  ( f `
 suc  n )  e.  _V
3836, 37opth1 4244 . . . . . . . . . 10  |-  ( <.
( f `  n
) ,  ( f `
 suc  n ) >.  =  <. 1o ,  1o >.  ->  ( f `  n )  =  1o )
3935, 38syl 15 . . . . . . . . 9  |-  ( <.
( f `  n
) ,  ( f `
 suc  n ) >.  e.  { <. 1o ,  1o >. }  ->  (
f `  n )  =  1o )
4034, 39sylbi 187 . . . . . . . 8  |-  ( ( f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n )  ->  (
f `  n )  =  1o )
41 tz6.12i 5548 . . . . . . . 8  |-  ( 1o  =/=  (/)  ->  ( (
f `  n )  =  1o  ->  n f 1o ) )
4233, 40, 41mpsyl 59 . . . . . . 7  |-  ( ( f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n )  ->  n
f 1o )
43 vex 2791 . . . . . . . 8  |-  n  e. 
_V
4443, 3breldm 4883 . . . . . . 7  |-  ( n f 1o  ->  n  e.  dom  f )
4542, 44syl 15 . . . . . 6  |-  ( ( f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n )  ->  n  e.  dom  f )
4645ralimi 2618 . . . . 5  |-  ( A. n  e.  om  (
f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n )  ->  A. n  e.  om  n  e.  dom  f )
47 dfss3 3170 . . . . 5  |-  ( om  C_  dom  f  <->  A. n  e.  om  n  e.  dom  f )
4846, 47sylibr 203 . . . 4  |-  ( A. n  e.  om  (
f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n )  ->  om  C_  dom  f )
49 vex 2791 . . . . . 6  |-  f  e. 
_V
5049dmex 4941 . . . . 5  |-  dom  f  e.  _V
5150ssex 4158 . . . 4  |-  ( om  C_  dom  f  ->  om  e.  _V )
5248, 51syl 15 . . 3  |-  ( A. n  e.  om  (
f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n )  ->  om  e.  _V )
5352exlimiv 1666 . 2  |-  ( E. f A. n  e. 
om  ( f `  n ) { <. 1o ,  1o >. }  (
f `  suc  n )  ->  om  e.  _V )
5432, 53ax-mp 8 1  |-  om  e.  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    C_ wss 3152   (/)c0 3455   {csn 3640   <.cop 3643   class class class wbr 4023   Oncon0 4392   suc csuc 4394   omcom 4656   dom cdm 4689   ran crn 4690   Fun wfun 5249   ` cfv 5255   1oc1o 6472
This theorem is referenced by:  axdc2lem  8074  axdc3lem  8076  axdc4lem  8081  axcclem  8083
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512  ax-dc 8072
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-1o 6479
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