MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cdalepw Unicode version

Theorem cdalepw 7909
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 5949 . . 3  |-  ( A  =  (/)  ->  ( A  +c  B )  =  ( (/)  +c  B
) )
21breq1d 4112 . 2  |-  ( A  =  (/)  ->  ( ( A  +c  B )  ~<_  ~P A  <->  ( (/)  +c  B
)  ~<_  ~P A ) )
3 relen 6953 . . . . . . . . 9  |-  Rel  ~~
43brrelex2i 4809 . . . . . . . 8  |-  ( ( A  +c  A ) 
~~  A  ->  A  e.  _V )
54adantr 451 . . . . . . 7  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  ->  A  e.  _V )
6 canth2g 7100 . . . . . . 7  |-  ( A  e.  _V  ->  A  ~<  ~P A )
7 sdomdom 6974 . . . . . . 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 7899 . . . . . . 7  |-  ( A  ~<_  ~P A  ->  ( A  +c  B )  ~<_  ( ~P A  +c  B
) )
11 cdadom2 7900 . . . . . . 7  |-  ( B  ~<_  ~P A  ->  ( ~P A  +c  B
)  ~<_  ( ~P A  +c  ~P A ) )
12 domtr 6999 . . . . . . 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 7907 . . . . . 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 7005 . . . . 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 7075 . . . . . . . . 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 7145 . . . . . . 7  |-  ( (/)  ~<  A 
<->  1o  ~<_  A )
2422, 23sylib 188 . . . . . 6  |-  ( ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  /\  A  =/=  (/) )  ->  1o  ~<_  A )
25 cdadom2 7900 . . . . . 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 7005 . . . . 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 7098 . . . 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 6999 . . 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 7894 . . 3  |-  ( (/)  +c  B )  ~~  ( B  +c  (/) )
35 reldom 6954 . . . . . . 7  |-  Rel  ~<_
3635brrelexi 4808 . . . . . 6  |-  ( B  ~<_  ~P A  ->  B  e.  _V )
3736adantl 452 . . . . 5  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  ->  B  e.  _V )
38 cda0en 7892 . . . . 5  |-  ( B  e.  _V  ->  ( B  +c  (/) )  ~~  B
)
39 domen1 7088 . . . . 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 7004 . . 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 2596 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 1642    e. wcel 1710    =/= wne 2521   _Vcvv 2864   (/)c0 3531   ~Pcpw 3701   class class class wbr 4102  (class class class)co 5942   1oc1o 6556    ~~ cen 6945    ~<_ cdom 6946    ~< csdm 6947    +c ccda 7880
This theorem is referenced by:  gchdomtri  8338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-1o 6563  df-2o 6564  df-er 6744  df-map 6859  df-en 6949  df-dom 6950  df-sdom 6951  df-cda 7881
  Copyright terms: Public domain W3C validator