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Theorem card2on 7998
Description: Proof that the alternate definition cardval2 8389 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 4912 . . . . . . . . . . . . 13  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  e.  On )
2 vex 3112 . . . . . . . . . . . . . 14  |-  z  e. 
_V
3 onelss 4929 . . . . . . . . . . . . . . 15  |-  ( z  e.  On  ->  (
y  e.  z  -> 
y  C_  z )
)
43imp 429 . . . . . . . . . . . . . 14  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  C_  z )
5 ssdomg 7580 . . . . . . . . . . . . . 14  |-  ( z  e.  _V  ->  (
y  C_  z  ->  y  ~<_  z ) )
62, 4, 5mpsyl 63 . . . . . . . . . . . . 13  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  ~<_  z )
71, 6jca 532 . . . . . . . . . . . 12  |-  ( ( z  e.  On  /\  y  e.  z )  ->  ( y  e.  On  /\  y  ~<_  z ) )
8 domsdomtr 7671 . . . . . . . . . . . . . 14  |-  ( ( y  ~<_  z  /\  z  ~<  A )  ->  y  ~<  A )
98anim2i 569 . . . . . . . . . . . . 13  |-  ( ( y  e.  On  /\  ( y  ~<_  z  /\  z  ~<  A ) )  ->  ( y  e.  On  /\  y  ~<  A ) )
109anassrs 648 . . . . . . . . . . . 12  |-  ( ( ( y  e.  On  /\  y  ~<_  z )  /\  z  ~<  A )  -> 
( y  e.  On  /\  y  ~<  A )
)
117, 10sylan 471 . . . . . . . . . . 11  |-  ( ( ( z  e.  On  /\  y  e.  z )  /\  z  ~<  A )  ->  ( y  e.  On  /\  y  ~<  A ) )
1211exp31 604 . . . . . . . . . 10  |-  ( z  e.  On  ->  (
y  e.  z  -> 
( z  ~<  A  -> 
( y  e.  On  /\  y  ~<  A )
) ) )
1312com12 31 . . . . . . . . 9  |-  ( y  e.  z  ->  (
z  e.  On  ->  ( z  ~<  A  ->  ( y  e.  On  /\  y  ~<  A ) ) ) )
1413impd 431 . . . . . . . 8  |-  ( y  e.  z  ->  (
( z  e.  On  /\  z  ~<  A )  ->  ( y  e.  On  /\  y  ~<  A )
) )
15 breq1 4459 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  ~<  A  <->  z  ~<  A ) )
1615elrab 3257 . . . . . . . 8  |-  ( z  e.  { x  e.  On  |  x  ~<  A }  <->  ( z  e.  On  /\  z  ~<  A ) )
17 breq1 4459 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  ~<  A  <->  y  ~<  A ) )
1817elrab 3257 . . . . . . . 8  |-  ( y  e.  { x  e.  On  |  x  ~<  A }  <->  ( y  e.  On  /\  y  ~<  A ) )
1914, 16, 183imtr4g 270 . . . . . . 7  |-  ( y  e.  z  ->  (
z  e.  { x  e.  On  |  x  ~<  A }  ->  y  e.  { x  e.  On  |  x  ~<  A } ) )
2019imp 429 . . . . . 6  |-  ( ( y  e.  z  /\  z  e.  { x  e.  On  |  x  ~<  A } )  ->  y  e.  { x  e.  On  |  x  ~<  A }
)
2120gen2 1620 . . . . 5  |-  A. y A. z ( ( y  e.  z  /\  z  e.  { x  e.  On  |  x  ~<  A }
)  ->  y  e.  { x  e.  On  |  x  ~<  A } )
22 dftr2 4552 . . . . 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 209 . . . 4  |-  Tr  {
x  e.  On  |  x  ~<  A }
24 ssrab2 3581 . . . 4  |-  { x  e.  On  |  x  ~<  A }  C_  On
25 ordon 6617 . . . 4  |-  Ord  On
26 trssord 4904 . . . 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 1324 . . 3  |-  Ord  {
x  e.  On  |  x  ~<  A }
28 hartogs 7987 . . . 4  |-  ( A  e.  _V  ->  { x  e.  On  |  x  ~<_  A }  e.  On )
29 sdomdom 7562 . . . . . . 7  |-  ( x 
~<  A  ->  x  ~<_  A )
3029a1i 11 . . . . . 6  |-  ( x  e.  On  ->  (
x  ~<  A  ->  x  ~<_  A ) )
3130ss2rabi 3578 . . . . 5  |-  { x  e.  On  |  x  ~<  A }  C_  { x  e.  On  |  x  ~<_  A }
32 ssexg 4602 . . . . 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 670 . . . 4  |-  ( { x  e.  On  |  x  ~<_  A }  e.  On  ->  { x  e.  On  |  x  ~<  A }  e.  _V )
34 elong 4895 . . . 4  |-  ( { x  e.  On  |  x  ~<  A }  e.  _V  ->  ( { x  e.  On  |  x  ~<  A }  e.  On  <->  Ord  { x  e.  On  |  x  ~<  A } ) )
3528, 33, 343syl 20 . . 3  |-  ( A  e.  _V  ->  ( { x  e.  On  |  x  ~<  A }  e.  On  <->  Ord  { x  e.  On  |  x  ~<  A } ) )
3627, 35mpbiri 233 . 2  |-  ( A  e.  _V  ->  { x  e.  On  |  x  ~<  A }  e.  On )
37 0elon 4940 . . . 4  |-  (/)  e.  On
38 eleq1 2529 . . . 4  |-  ( { x  e.  On  |  x  ~<  A }  =  (/) 
->  ( { x  e.  On  |  x  ~<  A }  e.  On  <->  (/)  e.  On ) )
3937, 38mpbiri 233 . . 3  |-  ( { x  e.  On  |  x  ~<  A }  =  (/) 
->  { x  e.  On  |  x  ~<  A }  e.  On )
40 df-ne 2654 . . . . 5  |-  ( { x  e.  On  |  x  ~<  A }  =/=  (/)  <->  -. 
{ x  e.  On  |  x  ~<  A }  =  (/) )
41 rabn0 3814 . . . . 5  |-  ( { x  e.  On  |  x  ~<  A }  =/=  (/)  <->  E. x  e.  On  x  ~<  A )
4240, 41bitr3i 251 . . . 4  |-  ( -. 
{ x  e.  On  |  x  ~<  A }  =  (/)  <->  E. x  e.  On  x  ~<  A )
43 relsdom 7542 . . . . . 6  |-  Rel  ~<
4443brrelex2i 5050 . . . . 5  |-  ( x 
~<  A  ->  A  e. 
_V )
4544rexlimivw 2946 . . . 4  |-  ( E. x  e.  On  x  ~<  A  ->  A  e.  _V )
4642, 45sylbi 195 . . 3  |-  ( -. 
{ x  e.  On  |  x  ~<  A }  =  (/)  ->  A  e.  _V )
4739, 46nsyl4 142 . 2  |-  ( -.  A  e.  _V  ->  { x  e.  On  |  x  ~<  A }  e.  On )
4836, 47pm2.61i 164 1  |-  { x  e.  On  |  x  ~<  A }  e.  On
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1393    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808   {crab 2811   _Vcvv 3109    C_ wss 3471   (/)c0 3793   class class class wbr 4456   Tr wtr 4550   Ord word 4886   Oncon0 4887    ~<_ cdom 7533    ~< csdm 7534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-recs 7060  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-oi 7953
This theorem is referenced by: (None)
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