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Theorem cdalepw 8032
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 6047 . . 3  |-  ( A  =  (/)  ->  ( A  +c  B )  =  ( (/)  +c  B
) )
21breq1d 4182 . 2  |-  ( A  =  (/)  ->  ( ( A  +c  B )  ~<_  ~P A  <->  ( (/)  +c  B
)  ~<_  ~P A ) )
3 relen 7073 . . . . . . . . 9  |-  Rel  ~~
43brrelex2i 4878 . . . . . . . 8  |-  ( ( A  +c  A ) 
~~  A  ->  A  e.  _V )
54adantr 452 . . . . . . 7  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  ->  A  e.  _V )
6 canth2g 7220 . . . . . . 7  |-  ( A  e.  _V  ->  A  ~<  ~P A )
7 sdomdom 7094 . . . . . . 7  |-  ( A 
~<  ~P A  ->  A  ~<_  ~P A )
85, 6, 73syl 19 . . . . . 6  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  ->  A  ~<_  ~P A )
9 simpr 448 . . . . . 6  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  ->  B  ~<_  ~P A )
10 cdadom1 8022 . . . . . . 7  |-  ( A  ~<_  ~P A  ->  ( A  +c  B )  ~<_  ( ~P A  +c  B
) )
11 cdadom2 8023 . . . . . . 7  |-  ( B  ~<_  ~P A  ->  ( ~P A  +c  B
)  ~<_  ( ~P A  +c  ~P A ) )
12 domtr 7119 . . . . . . 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 464 . . . . . 6  |-  ( ( A  ~<_  ~P A  /\  B  ~<_  ~P A )  ->  ( A  +c  B )  ~<_  ( ~P A  +c  ~P A ) )
148, 9, 13syl2anc 643 . . . . 5  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( A  +c  B
)  ~<_  ( ~P A  +c  ~P A ) )
15 pwcda1 8030 . . . . . 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 7125 . . . . 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 643 . . . 4  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( A  +c  B
)  ~<_  ~P ( A  +c  1o ) )
1918adantr 452 . . 3  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  ( A  +c  B )  ~<_  ~P ( A  +c  1o ) )
20 0sdomg 7195 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
215, 20syl 16 . . . . . . . 8  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( (/)  ~<  A  <->  A  =/=  (/) ) )
2221biimpar 472 . . . . . . 7  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  (/)  ~<  A )
23 0sdom1dom 7265 . . . . . . 7  |-  ( (/)  ~<  A 
<->  1o  ~<_  A )
2422, 23sylib 189 . . . . . 6  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  1o  ~<_  A )
25 cdadom2 8023 . . . . . 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 731 . . . . 5  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  ( A  +c  A )  ~~  A
)
28 domentr 7125 . . . . 5  |-  ( ( ( A  +c  1o )  ~<_  ( A  +c  A )  /\  ( A  +c  A )  ~~  A )  ->  ( A  +c  1o )  ~<_  A )
2926, 27, 28syl2anc 643 . . . 4  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  ( A  +c  1o )  ~<_  A )
30 pwdom 7218 . . . 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 7119 . . 3  |-  ( ( ( A  +c  B
)  ~<_  ~P ( A  +c  1o )  /\  ~P ( A  +c  1o )  ~<_  ~P A )  ->  ( A  +c  B )  ~<_  ~P A )
3319, 31, 32syl2anc 643 . 2  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  ( A  +c  B )  ~<_  ~P A
)
34 cdacomen 8017 . . 3  |-  ( (/)  +c  B )  ~~  ( B  +c  (/) )
35 reldom 7074 . . . . . . 7  |-  Rel  ~<_
3635brrelexi 4877 . . . . . 6  |-  ( B  ~<_  ~P A  ->  B  e.  _V )
3736adantl 453 . . . . 5  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  ->  B  e.  _V )
38 cda0en 8015 . . . . 5  |-  ( B  e.  _V  ->  ( B  +c  (/) )  ~~  B
)
39 domen1 7208 . . . . 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 224 . . 3  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( B  +c  (/) )  ~<_  ~P A )
42 endomtr 7124 . . 3  |-  ( ( ( (/)  +c  B
)  ~~  ( B  +c  (/) )  /\  ( B  +c  (/) )  ~<_  ~P A
)  ->  ( (/)  +c  B
)  ~<_  ~P A )
4334, 41, 42sylancr 645 . 2  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( (/)  +c  B )  ~<_  ~P A )
442, 33, 43pm2.61ne 2642 1  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( A  +c  B
)  ~<_  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916   (/)c0 3588   ~Pcpw 3759   class class class wbr 4172  (class class class)co 6040   1oc1o 6676    ~~ cen 7065    ~<_ cdom 7066    ~< csdm 7067    +c ccda 8003
This theorem is referenced by:  gchdomtri  8460
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 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-1o 6683  df-2o 6684  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-cda 8004
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