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Theorem cdalepw 7822
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 5865 . . 3  |-  ( A  =  (/)  ->  ( A  +c  B )  =  ( (/)  +c  B
) )
21breq1d 4033 . 2  |-  ( A  =  (/)  ->  ( ( A  +c  B )  ~<_  ~P A  <->  ( (/)  +c  B
)  ~<_  ~P A ) )
3 relen 6868 . . . . . . . . 9  |-  Rel  ~~
43brrelex2i 4730 . . . . . . . 8  |-  ( ( A  +c  A ) 
~~  A  ->  A  e.  _V )
54adantr 451 . . . . . . 7  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  ->  A  e.  _V )
6 canth2g 7015 . . . . . . 7  |-  ( A  e.  _V  ->  A  ~<  ~P A )
7 sdomdom 6889 . . . . . . 7  |-  ( A 
~<  ~P A  ->  A  ~<_  ~P A )
85, 6, 73syl 18 . . . . . 6  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  ->  A  ~<_  ~P A )
9 simpr 447 . . . . . 6  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  ->  B  ~<_  ~P A )
10 cdadom1 7812 . . . . . . 7  |-  ( A  ~<_  ~P A  ->  ( A  +c  B )  ~<_  ( ~P A  +c  B
) )
11 cdadom2 7813 . . . . . . 7  |-  ( B  ~<_  ~P A  ->  ( ~P A  +c  B
)  ~<_  ( ~P A  +c  ~P A ) )
12 domtr 6914 . . . . . . 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 463 . . . . . 6  |-  ( ( A  ~<_  ~P A  /\  B  ~<_  ~P A )  ->  ( A  +c  B )  ~<_  ( ~P A  +c  ~P A ) )
148, 9, 13syl2anc 642 . . . . 5  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( A  +c  B
)  ~<_  ( ~P A  +c  ~P A ) )
15 pwcda1 7820 . . . . . 6  |-  ( A  e.  _V  ->  ( ~P A  +c  ~P A
)  ~~  ~P ( A  +c  1o ) )
165, 15syl 15 . . . . 5  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( ~P A  +c  ~P A )  ~~  ~P ( A  +c  1o ) )
17 domentr 6920 . . . . 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 642 . . . 4  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( A  +c  B
)  ~<_  ~P ( A  +c  1o ) )
1918adantr 451 . . 3  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  ( A  +c  B )  ~<_  ~P ( A  +c  1o ) )
20 0sdomg 6990 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
215, 20syl 15 . . . . . . . 8  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( (/)  ~<  A  <->  A  =/=  (/) ) )
2221biimpar 471 . . . . . . 7  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  (/)  ~<  A )
23 0sdom1dom 7060 . . . . . . 7  |-  ( (/)  ~<  A 
<->  1o  ~<_  A )
2422, 23sylib 188 . . . . . 6  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  1o  ~<_  A )
25 cdadom2 7813 . . . . . 6  |-  ( 1o  ~<_  A  ->  ( A  +c  1o )  ~<_  ( A  +c  A ) )
2624, 25syl 15 . . . . 5  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  ( A  +c  1o )  ~<_  ( A  +c  A ) )
27 simpll 730 . . . . 5  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  ( A  +c  A )  ~~  A
)
28 domentr 6920 . . . . 5  |-  ( ( ( A  +c  1o )  ~<_  ( A  +c  A )  /\  ( A  +c  A )  ~~  A )  ->  ( A  +c  1o )  ~<_  A )
2926, 27, 28syl2anc 642 . . . 4  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  ( A  +c  1o )  ~<_  A )
30 pwdom 7013 . . . 4  |-  ( ( A  +c  1o )  ~<_  A  ->  ~P ( A  +c  1o )  ~<_  ~P A )
3129, 30syl 15 . . 3  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  ~P ( A  +c  1o )  ~<_  ~P A )
32 domtr 6914 . . 3  |-  ( ( ( A  +c  B
)  ~<_  ~P ( A  +c  1o )  /\  ~P ( A  +c  1o )  ~<_  ~P A )  ->  ( A  +c  B )  ~<_  ~P A )
3319, 31, 32syl2anc 642 . 2  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  ( A  +c  B )  ~<_  ~P A
)
34 cdacomen 7807 . . 3  |-  ( (/)  +c  B )  ~~  ( B  +c  (/) )
35 reldom 6869 . . . . . . 7  |-  Rel  ~<_
3635brrelexi 4729 . . . . . 6  |-  ( B  ~<_  ~P A  ->  B  e.  _V )
3736adantl 452 . . . . 5  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  ->  B  e.  _V )
38 cda0en 7805 . . . . 5  |-  ( B  e.  _V  ->  ( B  +c  (/) )  ~~  B
)
39 domen1 7003 . . . . 5  |-  ( ( B  +c  (/) )  ~~  B  ->  ( ( B  +c  (/) )  ~<_  ~P A  <->  B  ~<_  ~P A ) )
4037, 38, 393syl 18 . . . 4  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( ( B  +c  (/) )  ~<_  ~P A  <->  B  ~<_  ~P A
) )
419, 40mpbird 223 . . 3  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( B  +c  (/) )  ~<_  ~P A )
42 endomtr 6919 . . 3  |-  ( ( ( (/)  +c  B
)  ~~  ( B  +c  (/) )  /\  ( B  +c  (/) )  ~<_  ~P A
)  ->  ( (/)  +c  B
)  ~<_  ~P A )
4334, 41, 42sylancr 644 . 2  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( (/)  +c  B )  ~<_  ~P A )
442, 33, 43pm2.61ne 2521 1  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( A  +c  B
)  ~<_  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455   ~Pcpw 3625   class class class wbr 4023  (class class class)co 5858   1oc1o 6472    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862    +c ccda 7793
This theorem is referenced by:  gchdomtri  8251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-1o 6479  df-2o 6480  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-cda 7794
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