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Theorem cfslbn 8659
Description: Any subset of  A smaller than its cofinality has union less than  A. (This is the contrapositive to cfslb 8658.) (Contributed by Mario Carneiro, 24-Jun-2013.)
Hypothesis
Ref Expression
cfslb.1  |-  A  e. 
_V
Assertion
Ref Expression
cfslbn  |-  ( ( Lim  A  /\  B  C_  A  /\  B  ~<  ( cf `  A ) )  ->  U. B  e.  A )

Proof of Theorem cfslbn
StepHypRef Expression
1 uniss 4272 . . . . . . . 8  |-  ( B 
C_  A  ->  U. B  C_ 
U. A )
2 limuni 4944 . . . . . . . . 9  |-  ( Lim 
A  ->  A  =  U. A )
32sseq2d 3537 . . . . . . . 8  |-  ( Lim 
A  ->  ( U. B  C_  A  <->  U. B  C_  U. A ) )
41, 3syl5ibr 221 . . . . . . 7  |-  ( Lim 
A  ->  ( B  C_  A  ->  U. B  C_  A ) )
54imp 429 . . . . . 6  |-  ( ( Lim  A  /\  B  C_  A )  ->  U. B  C_  A )
6 limord 4943 . . . . . . . . . . . 12  |-  ( Lim 
A  ->  Ord  A )
7 ordsson 6620 . . . . . . . . . . . 12  |-  ( Ord 
A  ->  A  C_  On )
86, 7syl 16 . . . . . . . . . . 11  |-  ( Lim 
A  ->  A  C_  On )
9 sstr2 3516 . . . . . . . . . . 11  |-  ( B 
C_  A  ->  ( A  C_  On  ->  B  C_  On ) )
108, 9syl5com 30 . . . . . . . . . 10  |-  ( Lim 
A  ->  ( B  C_  A  ->  B  C_  On ) )
11 ssorduni 6616 . . . . . . . . . 10  |-  ( B 
C_  On  ->  Ord  U. B )
1210, 11syl6 33 . . . . . . . . 9  |-  ( Lim 
A  ->  ( B  C_  A  ->  Ord  U. B
) )
1312, 6jctird 544 . . . . . . . 8  |-  ( Lim 
A  ->  ( B  C_  A  ->  ( Ord  U. B  /\  Ord  A
) ) )
14 ordsseleq 4913 . . . . . . . 8  |-  ( ( Ord  U. B  /\  Ord  A )  ->  ( U. B  C_  A  <->  ( U. B  e.  A  \/  U. B  =  A ) ) )
1513, 14syl6 33 . . . . . . 7  |-  ( Lim 
A  ->  ( B  C_  A  ->  ( U. B  C_  A  <->  ( U. B  e.  A  \/  U. B  =  A ) ) ) )
1615imp 429 . . . . . 6  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( U. B  C_  A  <->  ( U. B  e.  A  \/  U. B  =  A ) ) )
175, 16mpbid 210 . . . . 5  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( U. B  e.  A  \/  U. B  =  A ) )
1817ord 377 . . . 4  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( -.  U. B  e.  A  ->  U. B  =  A ) )
19 cfslb.1 . . . . . . 7  |-  A  e. 
_V
2019cfslb 8658 . . . . . 6  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A )  ~<_  B )
21 domnsym 7655 . . . . . 6  |-  ( ( cf `  A )  ~<_  B  ->  -.  B  ~<  ( cf `  A
) )
2220, 21syl 16 . . . . 5  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  -.  B  ~<  ( cf `  A
) )
23223expia 1198 . . . 4  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( U. B  =  A  ->  -.  B  ~<  ( cf `  A ) ) )
2418, 23syld 44 . . 3  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( -.  U. B  e.  A  ->  -.  B  ~<  ( cf `  A ) ) )
2524con4d 105 . 2  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( B  ~<  ( cf `  A
)  ->  U. B  e.  A ) )
26253impia 1193 1  |-  ( ( Lim  A  /\  B  C_  A  /\  B  ~<  ( cf `  A ) )  ->  U. B  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3118    C_ wss 3481   U.cuni 4251   class class class wbr 4453   Ord word 4883   Oncon0 4884   Lim wlim 4885   ` cfv 5594    ~<_ cdom 7526    ~< csdm 7527   cfccf 8330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-recs 7054  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-card 8332  df-cf 8334
This theorem is referenced by:  cfslb2n  8660
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