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Theorem onacda 8616
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 7599 . . . . 5  |-  ( A  e.  On  ->  A  ~~  A )
21adantr 466 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  ~~  A )
3 simpr 462 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  B  e.  On )
4 eqid 2420 . . . . . . . 8  |-  ( x  e.  B  |->  ( A  +o  x ) )  =  ( x  e.  B  |->  ( A  +o  x ) )
54oacomf1olem 7264 . . . . . . 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 454 . . . . . 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 460 . . . . 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 7586 . . . . 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 665 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  B  ~~  ran  (
x  e.  B  |->  ( A  +o  x ) ) )
10 incom 3652 . . . . 5  |-  ( A  i^i  ran  ( x  e.  B  |->  ( A  +o  x ) ) )  =  ( ran  ( x  e.  B  |->  ( A  +o  x
) )  i^i  A
)
116simprd 464 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ran  ( x  e.  B  |->  ( A  +o  x ) )  i^i  A )  =  (/) )
1210, 11syl5eq 2473 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  i^i  ran  ( x  e.  B  |->  ( A  +o  x
) ) )  =  (/) )
13 cdaenun 8593 . . . 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 1264 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +c  B
)  ~~  ( A  u.  ran  ( x  e.  B  |->  ( A  +o  x ) ) ) )
15 oarec 7262 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( A  u.  ran  ( x  e.  B  |->  ( A  +o  x ) ) ) )
1614, 15breqtrrd 4443 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +c  B
)  ~~  ( A  +o  B ) )
1716ensymd 7618 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  ~~  ( A  +c  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867    u. cun 3431    i^i cin 3432   (/)c0 3758   class class class wbr 4417    |-> cmpt 4475   ran crn 4846   Oncon0 5433   -1-1-onto->wf1o 5591  (class class class)co 6296    +o coa 7178    ~~ cen 7565    +c ccda 8586
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 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
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 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  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-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-en 7569  df-cda 8587
This theorem is referenced by:  cardacda  8617  nnacda  8620
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