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Theorem dcomex 8825
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 4674 . . 3  |-  { <. 1o ,  1o >. }  e.  _V
2 1on 7135 . . . . . . . . . 10  |-  1o  e.  On
32elexi 3103 . . . . . . . . 9  |-  1o  e.  _V
43, 3fvsn 6085 . . . . . . . 8  |-  ( {
<. 1o ,  1o >. } `
 1o )  =  1o
53, 3funsn 5622 . . . . . . . . 9  |-  Fun  { <. 1o ,  1o >. }
63snid 4038 . . . . . . . . . 10  |-  1o  e.  { 1o }
73dmsnop 5468 . . . . . . . . . 10  |-  dom  { <. 1o ,  1o >. }  =  { 1o }
86, 7eleqtrri 2528 . . . . . . . . 9  |-  1o  e.  dom  { <. 1o ,  1o >. }
9 funbrfvb 5896 . . . . . . . . 9  |-  ( ( Fun  { <. 1o ,  1o >. }  /\  1o  e.  dom  { <. 1o ,  1o >. } )  -> 
( ( { <. 1o ,  1o >. } `  1o )  =  1o  <->  1o { <. 1o ,  1o >. } 1o ) )
105, 8, 9mp2an 672 . . . . . . . 8  |-  ( ( { <. 1o ,  1o >. } `  1o )  =  1o  <->  1o { <. 1o ,  1o >. } 1o )
114, 10mpbi 208 . . . . . . 7  |-  1o { <. 1o ,  1o >. } 1o
12 breq12 4438 . . . . . . . 8  |-  ( ( s  =  1o  /\  t  =  1o )  ->  ( s { <. 1o ,  1o >. } t  <-> 
1o { <. 1o ,  1o >. } 1o ) )
133, 3, 12spc2ev 3186 . . . . . . 7  |-  ( 1o { <. 1o ,  1o >. } 1o  ->  E. s E. t  s { <. 1o ,  1o >. } t )
1411, 13ax-mp 5 . . . . . 6  |-  E. s E. t  s { <. 1o ,  1o >. } t
15 breq 4435 . . . . . . 7  |-  ( x  =  { <. 1o ,  1o >. }  ->  (
s x t  <->  s { <. 1o ,  1o >. } t ) )
16152exbidv 1701 . . . . . 6  |-  ( x  =  { <. 1o ,  1o >. }  ->  ( E. s E. t  s x t  <->  E. s E. t  s { <. 1o ,  1o >. } t ) )
1714, 16mpbiri 233 . . . . 5  |-  ( x  =  { <. 1o ,  1o >. }  ->  E. s E. t  s x
t )
18 ssid 3505 . . . . . . 7  |-  { 1o }  C_  { 1o }
193rnsnop 5475 . . . . . . 7  |-  ran  { <. 1o ,  1o >. }  =  { 1o }
2018, 19, 73sstr4i 3525 . . . . . 6  |-  ran  { <. 1o ,  1o >. } 
C_  dom  { <. 1o ,  1o >. }
21 rneq 5214 . . . . . . 7  |-  ( x  =  { <. 1o ,  1o >. }  ->  ran  x  =  ran  { <. 1o ,  1o >. } )
22 dmeq 5189 . . . . . . 7  |-  ( x  =  { <. 1o ,  1o >. }  ->  dom  x  =  dom  { <. 1o ,  1o >. } )
2321, 22sseq12d 3515 . . . . . 6  |-  ( x  =  { <. 1o ,  1o >. }  ->  ( ran  x  C_  dom  x  <->  ran  { <. 1o ,  1o >. }  C_  dom  { <. 1o ,  1o >. } ) )
2420, 23mpbiri 233 . . . . 5  |-  ( x  =  { <. 1o ,  1o >. }  ->  ran  x  C_  dom  x )
25 pm5.5 336 . . . . 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 661 . . . 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 4435 . . . . . 6  |-  ( x  =  { <. 1o ,  1o >. }  ->  (
( f `  n
) x ( f `
 suc  n )  <->  ( f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n ) ) )
2827ralbidv 2880 . . . . 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 1699 . . . 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 253 . . 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 8824 . . 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 3145 . 2  |-  E. f A. n  e.  om  ( f `  n
) { <. 1o ,  1o >. }  ( f `
 suc  n )
33 1n0 7143 . . . . . . . 8  |-  1o  =/=  (/)
34 df-br 4434 . . . . . . . . 9  |-  ( ( f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n )  <->  <. ( f `
 n ) ,  ( f `  suc  n ) >.  e.  { <. 1o ,  1o >. } )
35 elsni 4035 . . . . . . . . . 10  |-  ( <.
( f `  n
) ,  ( f `
 suc  n ) >.  e.  { <. 1o ,  1o >. }  ->  <. (
f `  n ) ,  ( f `  suc  n ) >.  =  <. 1o ,  1o >. )
36 fvex 5862 . . . . . . . . . . 11  |-  ( f `
 n )  e. 
_V
37 fvex 5862 . . . . . . . . . . 11  |-  ( f `
 suc  n )  e.  _V
3836, 37opth1 4706 . . . . . . . . . 10  |-  ( <.
( f `  n
) ,  ( f `
 suc  n ) >.  =  <. 1o ,  1o >.  ->  ( f `  n )  =  1o )
3935, 38syl 16 . . . . . . . . 9  |-  ( <.
( f `  n
) ,  ( f `
 suc  n ) >.  e.  { <. 1o ,  1o >. }  ->  (
f `  n )  =  1o )
4034, 39sylbi 195 . . . . . . . 8  |-  ( ( f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n )  ->  (
f `  n )  =  1o )
41 tz6.12i 5872 . . . . . . . 8  |-  ( 1o  =/=  (/)  ->  ( (
f `  n )  =  1o  ->  n f 1o ) )
4233, 40, 41mpsyl 63 . . . . . . 7  |-  ( ( f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n )  ->  n
f 1o )
43 vex 3096 . . . . . . . 8  |-  n  e. 
_V
4443, 3breldm 5193 . . . . . . 7  |-  ( n f 1o  ->  n  e.  dom  f )
4542, 44syl 16 . . . . . 6  |-  ( ( f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n )  ->  n  e.  dom  f )
4645ralimi 2834 . . . . 5  |-  ( A. n  e.  om  (
f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n )  ->  A. n  e.  om  n  e.  dom  f )
47 dfss3 3476 . . . . 5  |-  ( om  C_  dom  f  <->  A. n  e.  om  n  e.  dom  f )
4846, 47sylibr 212 . . . 4  |-  ( A. n  e.  om  (
f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n )  ->  om  C_  dom  f )
49 vex 3096 . . . . . 6  |-  f  e. 
_V
5049dmex 6714 . . . . 5  |-  dom  f  e.  _V
5150ssex 4577 . . . 4  |-  ( om  C_  dom  f  ->  om  e.  _V )
5248, 51syl 16 . . 3  |-  ( A. n  e.  om  (
f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n )  ->  om  e.  _V )
5352exlimiv 1707 . 2  |-  ( E. f A. n  e. 
om  ( f `  n ) { <. 1o ,  1o >. }  (
f `  suc  n )  ->  om  e.  _V )
5432, 53ax-mp 5 1  |-  om  e.  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381   E.wex 1597    e. wcel 1802    =/= wne 2636   A.wral 2791   _Vcvv 3093    C_ wss 3458   (/)c0 3767   {csn 4010   <.cop 4016   class class class wbr 4433   Oncon0 4864   suc csuc 4866   dom cdm 4985   ran crn 4986   Fun wfun 5568   ` cfv 5574   omcom 6681   1oc1o 7121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672  ax-un 6573  ax-dc 8824
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-iota 5537  df-fun 5576  df-fn 5577  df-fv 5582  df-1o 7128
This theorem is referenced by:  axdc2lem  8826  axdc3lem  8828  axdc4lem  8833  axcclem  8835
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