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Theorem cdalepw 8107
Description: If  A is idempotent under cardinal sum and  B is dominated by the power set of  A, then so is the cardinal sum of  A and  B. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
cdalepw  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( A  +c  B
)  ~<_  ~P A )

Proof of Theorem cdalepw
StepHypRef Expression
1 oveq1 6117 . . 3  |-  ( A  =  (/)  ->  ( A  +c  B )  =  ( (/)  +c  B
) )
21breq1d 4247 . 2  |-  ( A  =  (/)  ->  ( ( A  +c  B )  ~<_  ~P A  <->  ( (/)  +c  B
)  ~<_  ~P A ) )
3 relen 7143 . . . . . . . . 9  |-  Rel  ~~
43brrelex2i 4948 . . . . . . . 8  |-  ( ( A  +c  A ) 
~~  A  ->  A  e.  _V )
54adantr 453 . . . . . . 7  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  ->  A  e.  _V )
6 canth2g 7290 . . . . . . 7  |-  ( A  e.  _V  ->  A  ~<  ~P A )
7 sdomdom 7164 . . . . . . 7  |-  ( A 
~<  ~P A  ->  A  ~<_  ~P A )
85, 6, 73syl 19 . . . . . 6  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  ->  A  ~<_  ~P A )
9 simpr 449 . . . . . 6  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  ->  B  ~<_  ~P A )
10 cdadom1 8097 . . . . . . 7  |-  ( A  ~<_  ~P A  ->  ( A  +c  B )  ~<_  ( ~P A  +c  B
) )
11 cdadom2 8098 . . . . . . 7  |-  ( B  ~<_  ~P A  ->  ( ~P A  +c  B
)  ~<_  ( ~P A  +c  ~P A ) )
12 domtr 7189 . . . . . . 7  |-  ( ( ( A  +c  B
)  ~<_  ( ~P A  +c  B )  /\  ( ~P A  +c  B
)  ~<_  ( ~P A  +c  ~P A ) )  ->  ( A  +c  B )  ~<_  ( ~P A  +c  ~P A
) )
1310, 11, 12syl2an 465 . . . . . 6  |-  ( ( A  ~<_  ~P A  /\  B  ~<_  ~P A )  ->  ( A  +c  B )  ~<_  ( ~P A  +c  ~P A ) )
148, 9, 13syl2anc 644 . . . . 5  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( A  +c  B
)  ~<_  ( ~P A  +c  ~P A ) )
15 pwcda1 8105 . . . . . 6  |-  ( A  e.  _V  ->  ( ~P A  +c  ~P A
)  ~~  ~P ( A  +c  1o ) )
165, 15syl 16 . . . . 5  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( ~P A  +c  ~P A )  ~~  ~P ( A  +c  1o ) )
17 domentr 7195 . . . . 5  |-  ( ( ( A  +c  B
)  ~<_  ( ~P A  +c  ~P A )  /\  ( ~P A  +c  ~P A )  ~~  ~P ( A  +c  1o ) )  ->  ( A  +c  B )  ~<_  ~P ( A  +c  1o ) )
1814, 16, 17syl2anc 644 . . . 4  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( A  +c  B
)  ~<_  ~P ( A  +c  1o ) )
1918adantr 453 . . 3  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  ( A  +c  B )  ~<_  ~P ( A  +c  1o ) )
20 0sdomg 7265 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
215, 20syl 16 . . . . . . . 8  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( (/)  ~<  A  <->  A  =/=  (/) ) )
2221biimpar 473 . . . . . . 7  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  (/)  ~<  A )
23 0sdom1dom 7335 . . . . . . 7  |-  ( (/)  ~<  A 
<->  1o  ~<_  A )
2422, 23sylib 190 . . . . . 6  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  1o  ~<_  A )
25 cdadom2 8098 . . . . . 6  |-  ( 1o  ~<_  A  ->  ( A  +c  1o )  ~<_  ( A  +c  A ) )
2624, 25syl 16 . . . . 5  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  ( A  +c  1o )  ~<_  ( A  +c  A ) )
27 simpll 732 . . . . 5  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  ( A  +c  A )  ~~  A
)
28 domentr 7195 . . . . 5  |-  ( ( ( A  +c  1o )  ~<_  ( A  +c  A )  /\  ( A  +c  A )  ~~  A )  ->  ( A  +c  1o )  ~<_  A )
2926, 27, 28syl2anc 644 . . . 4  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  ( A  +c  1o )  ~<_  A )
30 pwdom 7288 . . . 4  |-  ( ( A  +c  1o )  ~<_  A  ->  ~P ( A  +c  1o )  ~<_  ~P A )
3129, 30syl 16 . . 3  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  ~P ( A  +c  1o )  ~<_  ~P A )
32 domtr 7189 . . 3  |-  ( ( ( A  +c  B
)  ~<_  ~P ( A  +c  1o )  /\  ~P ( A  +c  1o )  ~<_  ~P A )  ->  ( A  +c  B )  ~<_  ~P A )
3319, 31, 32syl2anc 644 . 2  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  ( A  +c  B )  ~<_  ~P A
)
34 cdacomen 8092 . . 3  |-  ( (/)  +c  B )  ~~  ( B  +c  (/) )
35 reldom 7144 . . . . . . 7  |-  Rel  ~<_
3635brrelexi 4947 . . . . . 6  |-  ( B  ~<_  ~P A  ->  B  e.  _V )
3736adantl 454 . . . . 5  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  ->  B  e.  _V )
38 cda0en 8090 . . . . 5  |-  ( B  e.  _V  ->  ( B  +c  (/) )  ~~  B
)
39 domen1 7278 . . . . 5  |-  ( ( B  +c  (/) )  ~~  B  ->  ( ( B  +c  (/) )  ~<_  ~P A  <->  B  ~<_  ~P A ) )
4037, 38, 393syl 19 . . . 4  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( ( B  +c  (/) )  ~<_  ~P A  <->  B  ~<_  ~P A
) )
419, 40mpbird 225 . . 3  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( B  +c  (/) )  ~<_  ~P A )
42 endomtr 7194 . . 3  |-  ( ( ( (/)  +c  B
)  ~~  ( B  +c  (/) )  /\  ( B  +c  (/) )  ~<_  ~P A
)  ->  ( (/)  +c  B
)  ~<_  ~P A )
4334, 41, 42sylancr 646 . 2  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( (/)  +c  B )  ~<_  ~P A )
442, 33, 43pm2.61ne 2685 1  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( A  +c  B
)  ~<_  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727    =/= wne 2605   _Vcvv 2962   (/)c0 3613   ~Pcpw 3823   class class class wbr 4237  (class class class)co 6110   1oc1o 6746    ~~ cen 7135    ~<_ cdom 7136    ~< csdm 7137    +c ccda 8078
This theorem is referenced by:  gchdomtri  8535
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-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-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-1st 6378  df-2nd 6379  df-1o 6753  df-2o 6754  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-cda 8079
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