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Theorem unsnen 8191
Description: Equinumerosity of a set with a new element added. (Contributed by NM, 7-Nov-2008.)
Hypotheses
Ref Expression
unsnen.1  |-  A  e. 
_V
unsnen.2  |-  B  e. 
_V
Assertion
Ref Expression
unsnen  |-  ( -.  B  e.  A  -> 
( A  u.  { B } )  ~~  suc  ( card `  A )
)

Proof of Theorem unsnen
StepHypRef Expression
1 disjsn 3706 . . 3  |-  ( ( A  i^i  { B } )  =  (/)  <->  -.  B  e.  A )
2 cardon 7593 . . . . . 6  |-  ( card `  A )  e.  On
32onordi 4513 . . . . 5  |-  Ord  ( card `  A )
4 orddisj 4446 . . . . 5  |-  ( Ord  ( card `  A
)  ->  ( ( card `  A )  i^i 
{ ( card `  A
) } )  =  (/) )
53, 4ax-mp 8 . . . 4  |-  ( (
card `  A )  i^i  { ( card `  A
) } )  =  (/)
6 unsnen.1 . . . . . . 7  |-  A  e. 
_V
76cardid 8185 . . . . . 6  |-  ( card `  A )  ~~  A
87ensymi 6927 . . . . 5  |-  A  ~~  ( card `  A )
9 unsnen.2 . . . . . 6  |-  B  e. 
_V
10 fvex 5555 . . . . . 6  |-  ( card `  A )  e.  _V
11 en2sn 6956 . . . . . 6  |-  ( ( B  e.  _V  /\  ( card `  A )  e.  _V )  ->  { B }  ~~  { ( card `  A ) } )
129, 10, 11mp2an 653 . . . . 5  |-  { B }  ~~  { ( card `  A ) }
13 unen 6959 . . . . 5  |-  ( ( ( A  ~~  ( card `  A )  /\  { B }  ~~  {
( card `  A ) } )  /\  (
( A  i^i  { B } )  =  (/)  /\  ( ( card `  A
)  i^i  { ( card `  A ) } )  =  (/) ) )  ->  ( A  u.  { B } )  ~~  ( ( card `  A
)  u.  { (
card `  A ) } ) )
148, 12, 13mpanl12 663 . . . 4  |-  ( ( ( A  i^i  { B } )  =  (/)  /\  ( ( card `  A
)  i^i  { ( card `  A ) } )  =  (/) )  -> 
( A  u.  { B } )  ~~  (
( card `  A )  u.  { ( card `  A
) } ) )
155, 14mpan2 652 . . 3  |-  ( ( A  i^i  { B } )  =  (/)  ->  ( A  u.  { B } )  ~~  (
( card `  A )  u.  { ( card `  A
) } ) )
161, 15sylbir 204 . 2  |-  ( -.  B  e.  A  -> 
( A  u.  { B } )  ~~  (
( card `  A )  u.  { ( card `  A
) } ) )
17 df-suc 4414 . 2  |-  suc  ( card `  A )  =  ( ( card `  A
)  u.  { (
card `  A ) } )
1816, 17syl6breqr 4079 1  |-  ( -.  B  e.  A  -> 
( A  u.  { B } )  ~~  suc  ( card `  A )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163    i^i cin 3164   (/)c0 3468   {csn 3653   class class class wbr 4039   Ord word 4407   suc csuc 4410   ` cfv 5271    ~~ cen 6876   cardccrd 7584
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-ac2 8105
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-recs 6404  df-1o 6495  df-er 6676  df-en 6880  df-card 7588  df-ac 7759
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