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Theorem onacda 8108
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 7168 . . . . 5  |-  ( A  e.  On  ->  A  ~~  A )
21adantr 453 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  ~~  A )
3 simpr 449 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  B  e.  On )
4 eqid 2442 . . . . . . . 8  |-  ( x  e.  B  |->  ( A  +o  x ) )  =  ( x  e.  B  |->  ( A  +o  x ) )
54oacomf1olem 6836 . . . . . . 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 441 . . . . . 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 447 . . . . 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 7155 . . . . 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 644 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  B  ~~  ran  (
x  e.  B  |->  ( A  +o  x ) ) )
10 incom 3519 . . . . 5  |-  ( A  i^i  ran  ( x  e.  B  |->  ( A  +o  x ) ) )  =  ( ran  ( x  e.  B  |->  ( A  +o  x
) )  i^i  A
)
116simprd 451 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ran  ( x  e.  B  |->  ( A  +o  x ) )  i^i  A )  =  (/) )
1210, 11syl5eq 2486 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  i^i  ran  ( x  e.  B  |->  ( A  +o  x
) ) )  =  (/) )
13 cdaenun 8085 . . . 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 1185 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +c  B
)  ~~  ( A  u.  ran  ( x  e.  B  |->  ( A  +o  x ) ) ) )
15 oarec 6834 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( A  u.  ran  ( x  e.  B  |->  ( A  +o  x ) ) ) )
1614, 15breqtrrd 4263 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +c  B
)  ~~  ( A  +o  B ) )
1716ensymd 7187 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  ~~  ( A  +c  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727    u. cun 3304    i^i cin 3305   (/)c0 3613   class class class wbr 4237    e. cmpt 4291   Oncon0 4610   ran crn 4908   -1-1-onto->wf1o 5482  (class class class)co 6110    +o coa 6750    ~~ cen 7135    +c ccda 8078
This theorem is referenced by:  cardacda  8109  nnacda  8112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-er 6934  df-en 7139  df-cda 8079
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