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Theorem cfslbn 8139
Description: Any subset of  A smaller than its cofinality has union less than  A. (This is the contrapositive to cfslb 8138.) (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 4028 . . . . . . . 8  |-  ( B 
C_  A  ->  U. B  C_ 
U. A )
2 limuni 4633 . . . . . . . . 9  |-  ( Lim 
A  ->  A  =  U. A )
32sseq2d 3368 . . . . . . . 8  |-  ( Lim 
A  ->  ( U. B  C_  A  <->  U. B  C_  U. A ) )
41, 3syl5ibr 213 . . . . . . 7  |-  ( Lim 
A  ->  ( B  C_  A  ->  U. B  C_  A ) )
54imp 419 . . . . . 6  |-  ( ( Lim  A  /\  B  C_  A )  ->  U. B  C_  A )
6 limord 4632 . . . . . . . . . . . 12  |-  ( Lim 
A  ->  Ord  A )
7 ordsson 4762 . . . . . . . . . . . 12  |-  ( Ord 
A  ->  A  C_  On )
86, 7syl 16 . . . . . . . . . . 11  |-  ( Lim 
A  ->  A  C_  On )
9 sstr2 3347 . . . . . . . . . . 11  |-  ( B 
C_  A  ->  ( A  C_  On  ->  B  C_  On ) )
108, 9syl5com 28 . . . . . . . . . 10  |-  ( Lim 
A  ->  ( B  C_  A  ->  B  C_  On ) )
11 ssorduni 4758 . . . . . . . . . 10  |-  ( B 
C_  On  ->  Ord  U. B )
1210, 11syl6 31 . . . . . . . . 9  |-  ( Lim 
A  ->  ( B  C_  A  ->  Ord  U. B
) )
1312, 6jctird 529 . . . . . . . 8  |-  ( Lim 
A  ->  ( B  C_  A  ->  ( Ord  U. B  /\  Ord  A
) ) )
14 ordsseleq 4602 . . . . . . . 8  |-  ( ( Ord  U. B  /\  Ord  A )  ->  ( U. B  C_  A  <->  ( U. B  e.  A  \/  U. B  =  A ) ) )
1513, 14syl6 31 . . . . . . 7  |-  ( Lim 
A  ->  ( B  C_  A  ->  ( U. B  C_  A  <->  ( U. B  e.  A  \/  U. B  =  A ) ) ) )
1615imp 419 . . . . . 6  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( U. B  C_  A  <->  ( U. B  e.  A  \/  U. B  =  A ) ) )
175, 16mpbid 202 . . . . 5  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( U. B  e.  A  \/  U. B  =  A ) )
1817ord 367 . . . 4  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( -.  U. B  e.  A  ->  U. B  =  A ) )
19 cfslb.1 . . . . . . 7  |-  A  e. 
_V
2019cfslb 8138 . . . . . 6  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A )  ~<_  B )
21 domnsym 7225 . . . . . 6  |-  ( ( cf `  A )  ~<_  B  ->  -.  B  ~<  ( cf `  A
) )
2220, 21syl 16 . . . . 5  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  -.  B  ~<  ( cf `  A
) )
23223expia 1155 . . . 4  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( U. B  =  A  ->  -.  B  ~<  ( cf `  A ) ) )
2418, 23syld 42 . . 3  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( -.  U. B  e.  A  ->  -.  B  ~<  ( cf `  A ) ) )
2524con4d 99 . 2  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( B  ~<  ( cf `  A
)  ->  U. B  e.  A ) )
26253impia 1150 1  |-  ( ( Lim  A  /\  B  C_  A  /\  B  ~<  ( cf `  A ) )  ->  U. B  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   U.cuni 4007   class class class wbr 4204   Ord word 4572   Oncon0 4573   Lim wlim 4574   ` cfv 5446    ~<_ cdom 7099    ~< csdm 7100   cfccf 7816
This theorem is referenced by:  cfslb2n  8140
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
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  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-iin 4088  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-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-riota 6541  df-recs 6625  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-card 7818  df-cf 7820
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