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

Theorem onacda 8041
Description: The cardinal and ordinal sums are always equinumerous. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 30-May-2015.)
Assertion
Ref Expression
onacda  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  ~~  ( A  +c  B ) )

Proof of Theorem onacda
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 enrefg 7106 . . . . 5  |-  ( A  e.  On  ->  A  ~~  A )
21adantr 452 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  ~~  A )
3 simpr 448 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  B  e.  On )
4 eqid 2412 . . . . . . . 8  |-  ( x  e.  B  |->  ( A  +o  x ) )  =  ( x  e.  B  |->  ( A  +o  x ) )
54oacomf1olem 6774 . . . . . . 7  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( ( x  e.  B  |->  ( A  +o  x ) ) : B -1-1-onto-> ran  ( x  e.  B  |->  ( A  +o  x ) )  /\  ( ran  ( x  e.  B  |->  ( A  +o  x ) )  i^i 
A )  =  (/) ) )
65ancoms 440 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( x  e.  B  |->  ( A  +o  x ) ) : B -1-1-onto-> ran  ( x  e.  B  |->  ( A  +o  x ) )  /\  ( ran  ( x  e.  B  |->  ( A  +o  x ) )  i^i 
A )  =  (/) ) )
76simpld 446 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( x  e.  B  |->  ( A  +o  x
) ) : B -1-1-onto-> ran  ( x  e.  B  |->  ( A  +o  x
) ) )
8 f1oeng 7093 . . . . 5  |-  ( ( B  e.  On  /\  ( x  e.  B  |->  ( A  +o  x
) ) : B -1-1-onto-> ran  ( x  e.  B  |->  ( A  +o  x
) ) )  ->  B  ~~  ran  ( x  e.  B  |->  ( A  +o  x ) ) )
93, 7, 8syl2anc 643 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  B  ~~  ran  (
x  e.  B  |->  ( A  +o  x ) ) )
10 incom 3501 . . . . 5  |-  ( A  i^i  ran  ( x  e.  B  |->  ( A  +o  x ) ) )  =  ( ran  ( x  e.  B  |->  ( A  +o  x
) )  i^i  A
)
116simprd 450 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ran  ( x  e.  B  |->  ( A  +o  x ) )  i^i  A )  =  (/) )
1210, 11syl5eq 2456 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  i^i  ran  ( x  e.  B  |->  ( A  +o  x
) ) )  =  (/) )
13 cdaenun 8018 . . . 4  |-  ( ( A  ~~  A  /\  B  ~~  ran  ( x  e.  B  |->  ( A  +o  x ) )  /\  ( A  i^i  ran  ( x  e.  B  |->  ( A  +o  x
) ) )  =  (/) )  ->  ( A  +c  B )  ~~  ( A  u.  ran  ( x  e.  B  |->  ( A  +o  x
) ) ) )
142, 9, 12, 13syl3anc 1184 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +c  B
)  ~~  ( A  u.  ran  ( x  e.  B  |->  ( A  +o  x ) ) ) )
15 oarec 6772 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( A  u.  ran  ( x  e.  B  |->  ( A  +o  x ) ) ) )
1614, 15breqtrrd 4206 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +c  B
)  ~~  ( A  +o  B ) )
1716ensymd 7125 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  ~~  ( A  +c  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    u. cun 3286    i^i cin 3287   (/)c0 3596   class class class wbr 4180    e. cmpt 4234   Oncon0 4549   ran crn 4846   -1-1-onto->wf1o 5420  (class class class)co 6048    +o coa 6688    ~~ cen 7073    +c ccda 8011
This theorem is referenced by:  cardacda  8042  nnacda  8045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-cda 8012
  Copyright terms: Public domain W3C validator