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

Theorem gchcdaidm 8533
Description: An infinite GCH-set is idempotent under cardinal sum. Part of Lemma 2.2 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)
Assertion
Ref Expression
gchcdaidm  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~~  A )

Proof of Theorem gchcdaidm
StepHypRef Expression
1 simpl 444 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  e. GCH )
2 cdadom3 8058 . . . . 5  |-  ( ( A  e. GCH  /\  A  e. GCH )  ->  A  ~<_  ( A  +c  A ) )
31, 1, 2syl2anc 643 . . . 4  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<_  ( A  +c  A ) )
4 canth2g 7253 . . . . . . . . 9  |-  ( A  e. GCH  ->  A  ~<  ~P A
)
54adantr 452 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<  ~P A
)
6 sdomdom 7127 . . . . . . . 8  |-  ( A 
~<  ~P A  ->  A  ~<_  ~P A )
75, 6syl 16 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<_  ~P A )
8 cdadom1 8056 . . . . . . . 8  |-  ( A  ~<_  ~P A  ->  ( A  +c  A )  ~<_  ( ~P A  +c  A
) )
9 cdadom2 8057 . . . . . . . 8  |-  ( A  ~<_  ~P A  ->  ( ~P A  +c  A
)  ~<_  ( ~P A  +c  ~P A ) )
10 domtr 7152 . . . . . . . 8  |-  ( ( ( A  +c  A
)  ~<_  ( ~P A  +c  A )  /\  ( ~P A  +c  A
)  ~<_  ( ~P A  +c  ~P A ) )  ->  ( A  +c  A )  ~<_  ( ~P A  +c  ~P A
) )
118, 9, 10syl2anc 643 . . . . . . 7  |-  ( A  ~<_  ~P A  ->  ( A  +c  A )  ~<_  ( ~P A  +c  ~P A ) )
127, 11syl 16 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~<_  ( ~P A  +c  ~P A ) )
13 pwcda1 8064 . . . . . . . 8  |-  ( A  e. GCH  ->  ( ~P A  +c  ~P A )  ~~  ~P ( A  +c  1o ) )
1413adantr 452 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  +c  ~P A )  ~~  ~P ( A  +c  1o ) )
15 gchcda1 8521 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  1o )  ~~  A )
16 pwen 7272 . . . . . . . 8  |-  ( ( A  +c  1o ) 
~~  A  ->  ~P ( A  +c  1o )  ~~  ~P A )
1715, 16syl 16 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ~P ( A  +c  1o )  ~~  ~P A
)
18 entr 7151 . . . . . . 7  |-  ( ( ( ~P A  +c  ~P A )  ~~  ~P ( A  +c  1o )  /\  ~P ( A  +c  1o )  ~~  ~P A )  ->  ( ~P A  +c  ~P A
)  ~~  ~P A
)
1914, 17, 18syl2anc 643 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  +c  ~P A )  ~~  ~P A )
20 domentr 7158 . . . . . 6  |-  ( ( ( A  +c  A
)  ~<_  ( ~P A  +c  ~P A )  /\  ( ~P A  +c  ~P A )  ~~  ~P A )  ->  ( A  +c  A )  ~<_  ~P A )
2112, 19, 20syl2anc 643 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~<_  ~P A )
22 gchinf 8522 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  om  ~<_  A )
23 pwcdandom 8532 . . . . . . 7  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  +c  A
) )
2422, 23syl 16 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  -.  ~P A  ~<_  ( A  +c  A ) )
25 ensym 7148 . . . . . . 7  |-  ( ( A  +c  A ) 
~~  ~P A  ->  ~P A  ~~  ( A  +c  A ) )
26 endom 7126 . . . . . . 7  |-  ( ~P A  ~~  ( A  +c  A )  ->  ~P A  ~<_  ( A  +c  A ) )
2725, 26syl 16 . . . . . 6  |-  ( ( A  +c  A ) 
~~  ~P A  ->  ~P A  ~<_  ( A  +c  A ) )
2824, 27nsyl 115 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  -.  ( A  +c  A )  ~~  ~P A )
29 brsdom 7122 . . . . 5  |-  ( ( A  +c  A ) 
~<  ~P A  <->  ( ( A  +c  A )  ~<_  ~P A  /\  -.  ( A  +c  A )  ~~  ~P A ) )
3021, 28, 29sylanbrc 646 . . . 4  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~<  ~P A )
313, 30jca 519 . . 3  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  ~<_  ( A  +c  A )  /\  ( A  +c  A
)  ~<  ~P A ) )
32 gchen1 8490 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  ( A  +c  A )  /\  ( A  +c  A )  ~<  ~P A
) )  ->  A  ~~  ( A  +c  A
) )
3331, 32mpdan 650 . 2  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~~  ( A  +c  A ) )
3433ensymd 7150 1  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~~  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    e. wcel 1725   ~Pcpw 3791   class class class wbr 4204   omcom 4837  (class class class)co 6073   1oc1o 6709    ~~ cen 7098    ~<_ cdom 7099    ~< csdm 7100   Fincfn 7101    +c ccda 8037  GCHcgch 8485
This theorem is referenced by:  gchxpidm  8534  gchhar  8536  gchpwdom  8539
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7586
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-seqom 6697  df-1o 6716  df-2o 6717  df-oadd 6720  df-omul 6721  df-oexp 6722  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7469  df-har 7516  df-cnf 7607  df-card 7816  df-cda 8038  df-fin4 8157  df-gch 8486
  Copyright terms: Public domain W3C validator