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Theorem cdalepw 8040
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 6055 . . 3  |-  ( A  =  (/)  ->  ( A  +c  B )  =  ( (/)  +c  B
) )
21breq1d 4190 . 2  |-  ( A  =  (/)  ->  ( ( A  +c  B )  ~<_  ~P A  <->  ( (/)  +c  B
)  ~<_  ~P A ) )
3 relen 7081 . . . . . . . . 9  |-  Rel  ~~
43brrelex2i 4886 . . . . . . . 8  |-  ( ( A  +c  A ) 
~~  A  ->  A  e.  _V )
54adantr 452 . . . . . . 7  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  ->  A  e.  _V )
6 canth2g 7228 . . . . . . 7  |-  ( A  e.  _V  ->  A  ~<  ~P A )
7 sdomdom 7102 . . . . . . 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 8030 . . . . . . 7  |-  ( A  ~<_  ~P A  ->  ( A  +c  B )  ~<_  ( ~P A  +c  B
) )
11 cdadom2 8031 . . . . . . 7  |-  ( B  ~<_  ~P A  ->  ( ~P A  +c  B
)  ~<_  ( ~P A  +c  ~P A ) )
12 domtr 7127 . . . . . . 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 8038 . . . . . 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 7133 . . . . 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 7203 . . . . . . . . 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 7273 . . . . . . 7  |-  ( (/)  ~<  A 
<->  1o  ~<_  A )
2422, 23sylib 189 . . . . . 6  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  1o  ~<_  A )
25 cdadom2 8031 . . . . . 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 7133 . . . . 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 7226 . . . 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 7127 . . 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 8025 . . 3  |-  ( (/)  +c  B )  ~~  ( B  +c  (/) )
35 reldom 7082 . . . . . . 7  |-  Rel  ~<_
3635brrelexi 4885 . . . . . 6  |-  ( B  ~<_  ~P A  ->  B  e.  _V )
3736adantl 453 . . . . 5  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  ->  B  e.  _V )
38 cda0en 8023 . . . . 5  |-  ( B  e.  _V  ->  ( B  +c  (/) )  ~~  B
)
39 domen1 7216 . . . . 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 7132 . . 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 2650 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 2575   _Vcvv 2924   (/)c0 3596   ~Pcpw 3767   class class class wbr 4180  (class class class)co 6048   1oc1o 6684    ~~ cen 7073    ~<_ cdom 7074    ~< csdm 7075    +c ccda 8011
This theorem is referenced by:  gchdomtri  8468
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 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  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-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-1o 6691  df-2o 6692  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-cda 8012
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