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Theorem unsnen 8428
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 3868 . . 3  |-  ( ( A  i^i  { B } )  =  (/)  <->  -.  B  e.  A )
2 cardon 7831 . . . . . 6  |-  ( card `  A )  e.  On
32onordi 4686 . . . . 5  |-  Ord  ( card `  A )
4 orddisj 4619 . . . . 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 8422 . . . . . 6  |-  ( card `  A )  ~~  A
87ensymi 7157 . . . . 5  |-  A  ~~  ( card `  A )
9 unsnen.2 . . . . . 6  |-  B  e. 
_V
10 fvex 5742 . . . . . 6  |-  ( card `  A )  e.  _V
11 en2sn 7186 . . . . . 6  |-  ( ( B  e.  _V  /\  ( card `  A )  e.  _V )  ->  { B }  ~~  { ( card `  A ) } )
129, 10, 11mp2an 654 . . . . 5  |-  { B }  ~~  { ( card `  A ) }
13 unen 7189 . . . . 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 664 . . . 4  |-  ( ( ( A  i^i  { B } )  =  (/)  /\  ( ( card `  A
)  i^i  { ( card `  A ) } )  =  (/) )  -> 
( A  u.  { B } )  ~~  (
( card `  A )  u.  { ( card `  A
) } ) )
155, 14mpan2 653 . . 3  |-  ( ( A  i^i  { B } )  =  (/)  ->  ( A  u.  { B } )  ~~  (
( card `  A )  u.  { ( card `  A
) } ) )
161, 15sylbir 205 . 2  |-  ( -.  B  e.  A  -> 
( A  u.  { B } )  ~~  (
( card `  A )  u.  { ( card `  A
) } ) )
17 df-suc 4587 . 2  |-  suc  ( card `  A )  =  ( ( card `  A
)  u.  { (
card `  A ) } )
1816, 17syl6breqr 4252 1  |-  ( -.  B  e.  A  -> 
( A  u.  { B } )  ~~  suc  ( card `  A )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    u. cun 3318    i^i cin 3319   (/)c0 3628   {csn 3814   class class class wbr 4212   Ord word 4580   suc csuc 4583   ` cfv 5454    ~~ cen 7106   cardccrd 7822
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-ac2 8343
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-riota 6549  df-recs 6633  df-1o 6724  df-er 6905  df-en 7110  df-card 7826  df-ac 7997
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