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Theorem card2on 7523
Description: Proof that the alternate definition cardval2 7879 is always a set, and indeed is an ordinal. (Contributed by Mario Carneiro, 14-Jan-2013.)
Assertion
Ref Expression
card2on  |-  { x  e.  On  |  x  ~<  A }  e.  On
Distinct variable group:    x, A

Proof of Theorem card2on
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onelon 4607 . . . . . . . . . . . . 13  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  e.  On )
2 vex 2960 . . . . . . . . . . . . . 14  |-  z  e. 
_V
3 onelss 4624 . . . . . . . . . . . . . . 15  |-  ( z  e.  On  ->  (
y  e.  z  -> 
y  C_  z )
)
43imp 420 . . . . . . . . . . . . . 14  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  C_  z )
5 ssdomg 7154 . . . . . . . . . . . . . 14  |-  ( z  e.  _V  ->  (
y  C_  z  ->  y  ~<_  z ) )
62, 4, 5mpsyl 62 . . . . . . . . . . . . 13  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  ~<_  z )
71, 6jca 520 . . . . . . . . . . . 12  |-  ( ( z  e.  On  /\  y  e.  z )  ->  ( y  e.  On  /\  y  ~<_  z ) )
8 domsdomtr 7243 . . . . . . . . . . . . . 14  |-  ( ( y  ~<_  z  /\  z  ~<  A )  ->  y  ~<  A )
98anim2i 554 . . . . . . . . . . . . 13  |-  ( ( y  e.  On  /\  ( y  ~<_  z  /\  z  ~<  A ) )  ->  ( y  e.  On  /\  y  ~<  A ) )
109anassrs 631 . . . . . . . . . . . 12  |-  ( ( ( y  e.  On  /\  y  ~<_  z )  /\  z  ~<  A )  -> 
( y  e.  On  /\  y  ~<  A )
)
117, 10sylan 459 . . . . . . . . . . 11  |-  ( ( ( z  e.  On  /\  y  e.  z )  /\  z  ~<  A )  ->  ( y  e.  On  /\  y  ~<  A ) )
1211exp31 589 . . . . . . . . . 10  |-  ( z  e.  On  ->  (
y  e.  z  -> 
( z  ~<  A  -> 
( y  e.  On  /\  y  ~<  A )
) ) )
1312com12 30 . . . . . . . . 9  |-  ( y  e.  z  ->  (
z  e.  On  ->  ( z  ~<  A  ->  ( y  e.  On  /\  y  ~<  A ) ) ) )
1413imp3a 422 . . . . . . . 8  |-  ( y  e.  z  ->  (
( z  e.  On  /\  z  ~<  A )  ->  ( y  e.  On  /\  y  ~<  A )
) )
15 breq1 4216 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  ~<  A  <->  z  ~<  A ) )
1615elrab 3093 . . . . . . . 8  |-  ( z  e.  { x  e.  On  |  x  ~<  A }  <->  ( z  e.  On  /\  z  ~<  A ) )
17 breq1 4216 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  ~<  A  <->  y  ~<  A ) )
1817elrab 3093 . . . . . . . 8  |-  ( y  e.  { x  e.  On  |  x  ~<  A }  <->  ( y  e.  On  /\  y  ~<  A ) )
1914, 16, 183imtr4g 263 . . . . . . 7  |-  ( y  e.  z  ->  (
z  e.  { x  e.  On  |  x  ~<  A }  ->  y  e.  { x  e.  On  |  x  ~<  A } ) )
2019imp 420 . . . . . 6  |-  ( ( y  e.  z  /\  z  e.  { x  e.  On  |  x  ~<  A } )  ->  y  e.  { x  e.  On  |  x  ~<  A }
)
2120gen2 1557 . . . . 5  |-  A. y A. z ( ( y  e.  z  /\  z  e.  { x  e.  On  |  x  ~<  A }
)  ->  y  e.  { x  e.  On  |  x  ~<  A } )
22 dftr2 4305 . . . . 5  |-  ( Tr 
{ x  e.  On  |  x  ~<  A }  <->  A. y A. z ( ( y  e.  z  /\  z  e.  {
x  e.  On  |  x  ~<  A } )  ->  y  e.  {
x  e.  On  |  x  ~<  A } ) )
2321, 22mpbir 202 . . . 4  |-  Tr  {
x  e.  On  |  x  ~<  A }
24 ssrab2 3429 . . . 4  |-  { x  e.  On  |  x  ~<  A }  C_  On
25 ordon 4764 . . . 4  |-  Ord  On
26 trssord 4599 . . . 4  |-  ( ( Tr  { x  e.  On  |  x  ~<  A }  /\  { x  e.  On  |  x  ~<  A }  C_  On  /\  Ord  On )  ->  Ord  { x  e.  On  |  x  ~<  A } )
2723, 24, 25, 26mp3an 1280 . . 3  |-  Ord  {
x  e.  On  |  x  ~<  A }
28 hartogs 7514 . . . 4  |-  ( A  e.  _V  ->  { x  e.  On  |  x  ~<_  A }  e.  On )
29 sdomdom 7136 . . . . . . 7  |-  ( x 
~<  A  ->  x  ~<_  A )
3029a1i 11 . . . . . 6  |-  ( x  e.  On  ->  (
x  ~<  A  ->  x  ~<_  A ) )
3130ss2rabi 3426 . . . . 5  |-  { x  e.  On  |  x  ~<  A }  C_  { x  e.  On  |  x  ~<_  A }
32 ssexg 4350 . . . . 5  |-  ( ( { x  e.  On  |  x  ~<  A }  C_ 
{ x  e.  On  |  x  ~<_  A }  /\  { x  e.  On  |  x  ~<_  A }  e.  On )  ->  { x  e.  On  |  x  ~<  A }  e.  _V )
3331, 32mpan 653 . . . 4  |-  ( { x  e.  On  |  x  ~<_  A }  e.  On  ->  { x  e.  On  |  x  ~<  A }  e.  _V )
34 elong 4590 . . . 4  |-  ( { x  e.  On  |  x  ~<  A }  e.  _V  ->  ( { x  e.  On  |  x  ~<  A }  e.  On  <->  Ord  { x  e.  On  |  x  ~<  A } ) )
3528, 33, 343syl 19 . . 3  |-  ( A  e.  _V  ->  ( { x  e.  On  |  x  ~<  A }  e.  On  <->  Ord  { x  e.  On  |  x  ~<  A } ) )
3627, 35mpbiri 226 . 2  |-  ( A  e.  _V  ->  { x  e.  On  |  x  ~<  A }  e.  On )
37 0elon 4635 . . . 4  |-  (/)  e.  On
38 eleq1 2497 . . . 4  |-  ( { x  e.  On  |  x  ~<  A }  =  (/) 
->  ( { x  e.  On  |  x  ~<  A }  e.  On  <->  (/)  e.  On ) )
3937, 38mpbiri 226 . . 3  |-  ( { x  e.  On  |  x  ~<  A }  =  (/) 
->  { x  e.  On  |  x  ~<  A }  e.  On )
40 df-ne 2602 . . . . 5  |-  ( { x  e.  On  |  x  ~<  A }  =/=  (/)  <->  -. 
{ x  e.  On  |  x  ~<  A }  =  (/) )
41 rabn0 3648 . . . . 5  |-  ( { x  e.  On  |  x  ~<  A }  =/=  (/)  <->  E. x  e.  On  x  ~<  A )
4240, 41bitr3i 244 . . . 4  |-  ( -. 
{ x  e.  On  |  x  ~<  A }  =  (/)  <->  E. x  e.  On  x  ~<  A )
43 relsdom 7117 . . . . . 6  |-  Rel  ~<
4443brrelex2i 4920 . . . . 5  |-  ( x 
~<  A  ->  A  e. 
_V )
4544rexlimivw 2827 . . . 4  |-  ( E. x  e.  On  x  ~<  A  ->  A  e.  _V )
4642, 45sylbi 189 . . 3  |-  ( -. 
{ x  e.  On  |  x  ~<  A }  =  (/)  ->  A  e.  _V )
4739, 46nsyl4 137 . 2  |-  ( -.  A  e.  _V  ->  { x  e.  On  |  x  ~<  A }  e.  On )
4836, 47pm2.61i 159 1  |-  { x  e.  On  |  x  ~<  A }  e.  On
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726    =/= wne 2600   E.wrex 2707   {crab 2710   _Vcvv 2957    C_ wss 3321   (/)c0 3629   class class class wbr 4213   Tr wtr 4303   Ord word 4581   Oncon0 4582    ~<_ cdom 7108    ~< csdm 7109
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-riota 6550  df-recs 6634  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-oi 7480
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