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Theorem card2on 7486
Description: Proof that the alternate definition cardval2 7842 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 4574 . . . . . . . . . . . . 13  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  e.  On )
2 vex 2927 . . . . . . . . . . . . . 14  |-  z  e. 
_V
3 onelss 4591 . . . . . . . . . . . . . . 15  |-  ( z  e.  On  ->  (
y  e.  z  -> 
y  C_  z )
)
43imp 419 . . . . . . . . . . . . . 14  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  C_  z )
5 ssdomg 7120 . . . . . . . . . . . . . 14  |-  ( z  e.  _V  ->  (
y  C_  z  ->  y  ~<_  z ) )
62, 4, 5mpsyl 61 . . . . . . . . . . . . 13  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  ~<_  z )
71, 6jca 519 . . . . . . . . . . . 12  |-  ( ( z  e.  On  /\  y  e.  z )  ->  ( y  e.  On  /\  y  ~<_  z ) )
8 domsdomtr 7209 . . . . . . . . . . . . . 14  |-  ( ( y  ~<_  z  /\  z  ~<  A )  ->  y  ~<  A )
98anim2i 553 . . . . . . . . . . . . 13  |-  ( ( y  e.  On  /\  ( y  ~<_  z  /\  z  ~<  A ) )  ->  ( y  e.  On  /\  y  ~<  A ) )
109anassrs 630 . . . . . . . . . . . 12  |-  ( ( ( y  e.  On  /\  y  ~<_  z )  /\  z  ~<  A )  -> 
( y  e.  On  /\  y  ~<  A )
)
117, 10sylan 458 . . . . . . . . . . 11  |-  ( ( ( z  e.  On  /\  y  e.  z )  /\  z  ~<  A )  ->  ( y  e.  On  /\  y  ~<  A ) )
1211exp31 588 . . . . . . . . . 10  |-  ( z  e.  On  ->  (
y  e.  z  -> 
( z  ~<  A  -> 
( y  e.  On  /\  y  ~<  A )
) ) )
1312com12 29 . . . . . . . . 9  |-  ( y  e.  z  ->  (
z  e.  On  ->  ( z  ~<  A  ->  ( y  e.  On  /\  y  ~<  A ) ) ) )
1413imp3a 421 . . . . . . . 8  |-  ( y  e.  z  ->  (
( z  e.  On  /\  z  ~<  A )  ->  ( y  e.  On  /\  y  ~<  A )
) )
15 breq1 4183 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  ~<  A  <->  z  ~<  A ) )
1615elrab 3060 . . . . . . . 8  |-  ( z  e.  { x  e.  On  |  x  ~<  A }  <->  ( z  e.  On  /\  z  ~<  A ) )
17 breq1 4183 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  ~<  A  <->  y  ~<  A ) )
1817elrab 3060 . . . . . . . 8  |-  ( y  e.  { x  e.  On  |  x  ~<  A }  <->  ( y  e.  On  /\  y  ~<  A ) )
1914, 16, 183imtr4g 262 . . . . . . 7  |-  ( y  e.  z  ->  (
z  e.  { x  e.  On  |  x  ~<  A }  ->  y  e.  { x  e.  On  |  x  ~<  A } ) )
2019imp 419 . . . . . 6  |-  ( ( y  e.  z  /\  z  e.  { x  e.  On  |  x  ~<  A } )  ->  y  e.  { x  e.  On  |  x  ~<  A }
)
2120gen2 1553 . . . . 5  |-  A. y A. z ( ( y  e.  z  /\  z  e.  { x  e.  On  |  x  ~<  A }
)  ->  y  e.  { x  e.  On  |  x  ~<  A } )
22 dftr2 4272 . . . . 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 201 . . . 4  |-  Tr  {
x  e.  On  |  x  ~<  A }
24 ssrab2 3396 . . . 4  |-  { x  e.  On  |  x  ~<  A }  C_  On
25 ordon 4730 . . . 4  |-  Ord  On
26 trssord 4566 . . . 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 1279 . . 3  |-  Ord  {
x  e.  On  |  x  ~<  A }
28 hartogs 7477 . . . 4  |-  ( A  e.  _V  ->  { x  e.  On  |  x  ~<_  A }  e.  On )
29 sdomdom 7102 . . . . . . 7  |-  ( x 
~<  A  ->  x  ~<_  A )
3029a1i 11 . . . . . 6  |-  ( x  e.  On  ->  (
x  ~<  A  ->  x  ~<_  A ) )
3130ss2rabi 3393 . . . . 5  |-  { x  e.  On  |  x  ~<  A }  C_  { x  e.  On  |  x  ~<_  A }
32 ssexg 4317 . . . . 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 652 . . . 4  |-  ( { x  e.  On  |  x  ~<_  A }  e.  On  ->  { x  e.  On  |  x  ~<  A }  e.  _V )
34 elong 4557 . . . 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 225 . 2  |-  ( A  e.  _V  ->  { x  e.  On  |  x  ~<  A }  e.  On )
37 0elon 4602 . . . 4  |-  (/)  e.  On
38 eleq1 2472 . . . 4  |-  ( { x  e.  On  |  x  ~<  A }  =  (/) 
->  ( { x  e.  On  |  x  ~<  A }  e.  On  <->  (/)  e.  On ) )
3937, 38mpbiri 225 . . 3  |-  ( { x  e.  On  |  x  ~<  A }  =  (/) 
->  { x  e.  On  |  x  ~<  A }  e.  On )
40 df-ne 2577 . . . . 5  |-  ( { x  e.  On  |  x  ~<  A }  =/=  (/)  <->  -. 
{ x  e.  On  |  x  ~<  A }  =  (/) )
41 rabn0 3615 . . . . 5  |-  ( { x  e.  On  |  x  ~<  A }  =/=  (/)  <->  E. x  e.  On  x  ~<  A )
4240, 41bitr3i 243 . . . 4  |-  ( -. 
{ x  e.  On  |  x  ~<  A }  =  (/)  <->  E. x  e.  On  x  ~<  A )
43 relsdom 7083 . . . . . 6  |-  Rel  ~<
4443brrelex2i 4886 . . . . 5  |-  ( x 
~<  A  ->  A  e. 
_V )
4544rexlimivw 2794 . . . 4  |-  ( E. x  e.  On  x  ~<  A  ->  A  e.  _V )
4642, 45sylbi 188 . . 3  |-  ( -. 
{ x  e.  On  |  x  ~<  A }  =  (/)  ->  A  e.  _V )
4739, 46nsyl4 136 . 2  |-  ( -.  A  e.  _V  ->  { x  e.  On  |  x  ~<  A }  e.  On )
4836, 47pm2.61i 158 1  |-  { x  e.  On  |  x  ~<  A }  e.  On
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1721    =/= wne 2575   E.wrex 2675   {crab 2678   _Vcvv 2924    C_ wss 3288   (/)c0 3596   class class class wbr 4180   Tr wtr 4270   Ord word 4548   Oncon0 4549    ~<_ cdom 7074    ~< csdm 7075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6516  df-recs 6600  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-oi 7443
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