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Theorem inawina 8570
Description: Every strongly inaccessible cardinal is weakly inaccessible. (Contributed by Mario Carneiro, 29-May-2014.)
Assertion
Ref Expression
inawina  |-  ( A  e.  Inacc  ->  A  e.  Inacc W )

Proof of Theorem inawina
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfon 8140 . . . . 5  |-  ( cf `  A )  e.  On
2 eleq1 2498 . . . . 5  |-  ( ( cf `  A )  =  A  ->  (
( cf `  A
)  e.  On  <->  A  e.  On ) )
31, 2mpbii 204 . . . 4  |-  ( ( cf `  A )  =  A  ->  A  e.  On )
433ad2ant2 980 . . 3  |-  ( ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A )  ->  A  e.  On )
5 idd 23 . . . 4  |-  ( A  e.  On  ->  ( A  =/=  (/)  ->  A  =/=  (/) ) )
6 idd 23 . . . 4  |-  ( A  e.  On  ->  (
( cf `  A
)  =  A  -> 
( cf `  A
)  =  A ) )
7 inawinalem 8569 . . . 4  |-  ( A  e.  On  ->  ( A. x  e.  A  ~P x  ~<  A  ->  A. x  e.  A  E. y  e.  A  x  ~<  y ) )
85, 6, 73anim123d 1262 . . 3  |-  ( A  e.  On  ->  (
( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A )  ->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y
) ) )
94, 8mpcom 35 . 2  |-  ( ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A )  ->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y
) )
10 elina 8567 . 2  |-  ( A  e.  Inacc 
<->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A ) )
11 elwina 8566 . 2  |-  ( A  e.  Inacc W  <->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y
) )
129, 10, 113imtr4i 259 1  |-  ( A  e.  Inacc  ->  A  e.  Inacc W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   (/)c0 3630   ~Pcpw 3801   class class class wbr 4215   Oncon0 4584   ` cfv 5457    ~< csdm 7111   cfccf 7829   Inacc Wcwina 8562   Inacccina 8563
This theorem is referenced by:  gchina  8579  inar1  8655  inatsk  8658  tskuni  8663  grur1a  8699  grur1  8700  inaprc  8716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-suc 4590  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-riota 6552  df-recs 6636  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-card 7831  df-cf 7833  df-wina 8564  df-ina 8565
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