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Theorem dcomex 8612
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 4530 . . 3  |-  { <. 1o ,  1o >. }  e.  _V
2 1on 6923 . . . . . . . . . 10  |-  1o  e.  On
32elexi 2980 . . . . . . . . 9  |-  1o  e.  _V
43, 3fvsn 5908 . . . . . . . 8  |-  ( {
<. 1o ,  1o >. } `
 1o )  =  1o
53, 3funsn 5463 . . . . . . . . 9  |-  Fun  { <. 1o ,  1o >. }
63snid 3902 . . . . . . . . . 10  |-  1o  e.  { 1o }
73dmsnop 5310 . . . . . . . . . 10  |-  dom  { <. 1o ,  1o >. }  =  { 1o }
86, 7eleqtrri 2514 . . . . . . . . 9  |-  1o  e.  dom  { <. 1o ,  1o >. }
9 funbrfvb 5731 . . . . . . . . 9  |-  ( ( Fun  { <. 1o ,  1o >. }  /\  1o  e.  dom  { <. 1o ,  1o >. } )  -> 
( ( { <. 1o ,  1o >. } `  1o )  =  1o  <->  1o { <. 1o ,  1o >. } 1o ) )
105, 8, 9mp2an 667 . . . . . . . 8  |-  ( ( { <. 1o ,  1o >. } `  1o )  =  1o  <->  1o { <. 1o ,  1o >. } 1o )
114, 10mpbi 208 . . . . . . 7  |-  1o { <. 1o ,  1o >. } 1o
12 breq12 4294 . . . . . . . 8  |-  ( ( s  =  1o  /\  t  =  1o )  ->  ( s { <. 1o ,  1o >. } t  <-> 
1o { <. 1o ,  1o >. } 1o ) )
133, 3, 12spc2ev 3062 . . . . . . 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 4291 . . . . . . 7  |-  ( x  =  { <. 1o ,  1o >. }  ->  (
s x t  <->  s { <. 1o ,  1o >. } t ) )
16152exbidv 1687 . . . . . 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 3372 . . . . . . 7  |-  { 1o }  C_  { 1o }
193rnsnop 5317 . . . . . . 7  |-  ran  { <. 1o ,  1o >. }  =  { 1o }
2018, 19, 73sstr4i 3392 . . . . . 6  |-  ran  { <. 1o ,  1o >. } 
C_  dom  { <. 1o ,  1o >. }
21 rneq 5061 . . . . . . 7  |-  ( x  =  { <. 1o ,  1o >. }  ->  ran  x  =  ran  { <. 1o ,  1o >. } )
22 dmeq 5036 . . . . . . 7  |-  ( x  =  { <. 1o ,  1o >. }  ->  dom  x  =  dom  { <. 1o ,  1o >. } )
2321, 22sseq12d 3382 . . . . . 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 656 . . . 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 4291 . . . . . 6  |-  ( x  =  { <. 1o ,  1o >. }  ->  (
( f `  n
) x ( f `
 suc  n )  <->  ( f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n ) ) )
2827ralbidv 2733 . . . . 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 1685 . . . 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 8611 . . 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 3021 . 2  |-  E. f A. n  e.  om  ( f `  n
) { <. 1o ,  1o >. }  ( f `
 suc  n )
33 1n0 6931 . . . . . . . 8  |-  1o  =/=  (/)
34 df-br 4290 . . . . . . . . 9  |-  ( ( f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n )  <->  <. ( f `
 n ) ,  ( f `  suc  n ) >.  e.  { <. 1o ,  1o >. } )
35 elsni 3899 . . . . . . . . . 10  |-  ( <.
( f `  n
) ,  ( f `
 suc  n ) >.  e.  { <. 1o ,  1o >. }  ->  <. (
f `  n ) ,  ( f `  suc  n ) >.  =  <. 1o ,  1o >. )
36 fvex 5698 . . . . . . . . . . 11  |-  ( f `
 n )  e. 
_V
37 fvex 5698 . . . . . . . . . . 11  |-  ( f `
 suc  n )  e.  _V
3836, 37opth1 4562 . . . . . . . . . 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 5707 . . . . . . . 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 2973 . . . . . . . 8  |-  n  e. 
_V
4443, 3breldm 5040 . . . . . . 7  |-  ( n f 1o  ->  n  e.  dom  f )
4542, 44syl 16 . . . . . 6  |-  ( ( f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n )  ->  n  e.  dom  f )
4645ralimi 2789 . . . . 5  |-  ( A. n  e.  om  (
f `  n ) { <. 1o ,  1o >. }  ( f `  suc  n )  ->  A. n  e.  om  n  e.  dom  f )
47 dfss3 3343 . . . . 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 2973 . . . . . 6  |-  f  e. 
_V
5049dmex 6510 . . . . 5  |-  dom  f  e.  _V
5150ssex 4433 . . . 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 1693 . 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 1364   E.wex 1591    e. wcel 1761    =/= wne 2604   A.wral 2713   _Vcvv 2970    C_ wss 3325   (/)c0 3634   {csn 3874   <.cop 3880   class class class wbr 4289   Oncon0 4715   suc csuc 4717   dom cdm 4836   ran crn 4837   Fun wfun 5409   ` cfv 5415   omcom 6475   1oc1o 6909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371  ax-dc 8611
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-iota 5378  df-fun 5417  df-fn 5418  df-fv 5423  df-1o 6916
This theorem is referenced by:  axdc2lem  8613  axdc3lem  8615  axdc4lem  8620  axcclem  8622
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