MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unsnen Structured version   Unicode version

Theorem unsnen 8979
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 4057 . . 3  |-  ( ( A  i^i  { B } )  =  (/)  <->  -.  B  e.  A )
2 cardon 8380 . . . . . 6  |-  ( card `  A )  e.  On
32onordi 5543 . . . . 5  |-  Ord  ( card `  A )
4 orddisj 5477 . . . . 5  |-  ( Ord  ( card `  A
)  ->  ( ( card `  A )  i^i 
{ ( card `  A
) } )  =  (/) )
53, 4ax-mp 5 . . . 4  |-  ( (
card `  A )  i^i  { ( card `  A
) } )  =  (/)
6 unsnen.1 . . . . . . 7  |-  A  e. 
_V
76cardid 8973 . . . . . 6  |-  ( card `  A )  ~~  A
87ensymi 7623 . . . . 5  |-  A  ~~  ( card `  A )
9 unsnen.2 . . . . . 6  |-  B  e. 
_V
10 fvex 5888 . . . . . 6  |-  ( card `  A )  e.  _V
11 en2sn 7653 . . . . . 6  |-  ( ( B  e.  _V  /\  ( card `  A )  e.  _V )  ->  { B }  ~~  { ( card `  A ) } )
129, 10, 11mp2an 676 . . . . 5  |-  { B }  ~~  { ( card `  A ) }
13 unen 7656 . . . . 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 686 . . . 4  |-  ( ( ( A  i^i  { B } )  =  (/)  /\  ( ( card `  A
)  i^i  { ( card `  A ) } )  =  (/) )  -> 
( A  u.  { B } )  ~~  (
( card `  A )  u.  { ( card `  A
) } ) )
155, 14mpan2 675 . . 3  |-  ( ( A  i^i  { B } )  =  (/)  ->  ( A  u.  { B } )  ~~  (
( card `  A )  u.  { ( card `  A
) } ) )
161, 15sylbir 216 . 2  |-  ( -.  B  e.  A  -> 
( A  u.  { B } )  ~~  (
( card `  A )  u.  { ( card `  A
) } ) )
17 df-suc 5445 . 2  |-  suc  ( card `  A )  =  ( ( card `  A
)  u.  { (
card `  A ) } )
1816, 17syl6breqr 4461 1  |-  ( -.  B  e.  A  -> 
( A  u.  { B } )  ~~  suc  ( card `  A )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868   _Vcvv 3081    u. cun 3434    i^i cin 3435   (/)c0 3761   {csn 3996   class class class wbr 4420   Ord word 5438   suc csuc 5441   ` cfv 5598    ~~ cen 7571   cardccrd 8371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-ac2 8894
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-wrecs 7033  df-recs 7095  df-1o 7187  df-er 7368  df-en 7575  df-card 8375  df-ac 8548
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator