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Theorem unsnen 8175
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 3693 . . 3  |-  ( ( A  i^i  { B } )  =  (/)  <->  -.  B  e.  A )
2 cardon 7577 . . . . . 6  |-  ( card `  A )  e.  On
32onordi 4497 . . . . 5  |-  Ord  ( card `  A )
4 orddisj 4430 . . . . 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 8169 . . . . . 6  |-  ( card `  A )  ~~  A
87ensymi 6911 . . . . 5  |-  A  ~~  ( card `  A )
9 unsnen.2 . . . . . 6  |-  B  e. 
_V
10 fvex 5539 . . . . . 6  |-  ( card `  A )  e.  _V
11 en2sn 6940 . . . . . 6  |-  ( ( B  e.  _V  /\  ( card `  A )  e.  _V )  ->  { B }  ~~  { ( card `  A ) } )
129, 10, 11mp2an 653 . . . . 5  |-  { B }  ~~  { ( card `  A ) }
13 unen 6943 . . . . 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 4398 . 2  |-  suc  ( card `  A )  =  ( ( card `  A
)  u.  { (
card `  A ) } )
1816, 17syl6breqr 4063 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 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150    i^i cin 3151   (/)c0 3455   {csn 3640   class class class wbr 4023   Ord word 4391   suc csuc 4394   ` cfv 5255    ~~ cen 6860   cardccrd 7568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-ac2 8089
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-1o 6479  df-er 6660  df-en 6864  df-card 7572  df-ac 7743
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