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Theorem onacda 8578
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 7548 . . . . 5  |-  ( A  e.  On  ->  A  ~~  A )
21adantr 465 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  ~~  A )
3 simpr 461 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  B  e.  On )
4 eqid 2467 . . . . . . . 8  |-  ( x  e.  B  |->  ( A  +o  x ) )  =  ( x  e.  B  |->  ( A  +o  x ) )
54oacomf1olem 7214 . . . . . . 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 453 . . . . . 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 459 . . . . 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 7535 . . . . 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 661 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  B  ~~  ran  (
x  e.  B  |->  ( A  +o  x ) ) )
10 incom 3691 . . . . 5  |-  ( A  i^i  ran  ( x  e.  B  |->  ( A  +o  x ) ) )  =  ( ran  ( x  e.  B  |->  ( A  +o  x
) )  i^i  A
)
116simprd 463 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ran  ( x  e.  B  |->  ( A  +o  x ) )  i^i  A )  =  (/) )
1210, 11syl5eq 2520 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  i^i  ran  ( x  e.  B  |->  ( A  +o  x
) ) )  =  (/) )
13 cdaenun 8555 . . . 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 1228 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +c  B
)  ~~  ( A  u.  ran  ( x  e.  B  |->  ( A  +o  x ) ) ) )
15 oarec 7212 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( A  u.  ran  ( x  e.  B  |->  ( A  +o  x ) ) ) )
1614, 15breqtrrd 4473 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +c  B
)  ~~  ( A  +o  B ) )
1716ensymd 7567 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  ~~  ( A  +c  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    u. cun 3474    i^i cin 3475   (/)c0 3785   class class class wbr 4447    |-> cmpt 4505   Oncon0 4878   ran crn 5000   -1-1-onto->wf1o 5587  (class class class)co 6285    +o coa 7128    ~~ cen 7514    +c ccda 8548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-en 7518  df-cda 8549
This theorem is referenced by:  cardacda  8579  nnacda  8582
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