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Theorem cfslbn 7909
Description: Any subset of  A smaller than its cofinality has union less than  A. (This is the contrapositive to cfslb 7908.) (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 3864 . . . . . . . 8  |-  ( B 
C_  A  ->  U. B  C_ 
U. A )
2 limuni 4468 . . . . . . . . 9  |-  ( Lim 
A  ->  A  =  U. A )
32sseq2d 3219 . . . . . . . 8  |-  ( Lim 
A  ->  ( U. B  C_  A  <->  U. B  C_  U. A ) )
41, 3syl5ibr 212 . . . . . . 7  |-  ( Lim 
A  ->  ( B  C_  A  ->  U. B  C_  A ) )
54imp 418 . . . . . 6  |-  ( ( Lim  A  /\  B  C_  A )  ->  U. B  C_  A )
6 limord 4467 . . . . . . . . . . . 12  |-  ( Lim 
A  ->  Ord  A )
7 ordsson 4597 . . . . . . . . . . . 12  |-  ( Ord 
A  ->  A  C_  On )
86, 7syl 15 . . . . . . . . . . 11  |-  ( Lim 
A  ->  A  C_  On )
9 sstr2 3199 . . . . . . . . . . 11  |-  ( B 
C_  A  ->  ( A  C_  On  ->  B  C_  On ) )
108, 9syl5com 26 . . . . . . . . . 10  |-  ( Lim 
A  ->  ( B  C_  A  ->  B  C_  On ) )
11 ssorduni 4593 . . . . . . . . . 10  |-  ( B 
C_  On  ->  Ord  U. B )
1210, 11syl6 29 . . . . . . . . 9  |-  ( Lim 
A  ->  ( B  C_  A  ->  Ord  U. B
) )
1312, 6jctird 528 . . . . . . . 8  |-  ( Lim 
A  ->  ( B  C_  A  ->  ( Ord  U. B  /\  Ord  A
) ) )
14 ordsseleq 4437 . . . . . . . 8  |-  ( ( Ord  U. B  /\  Ord  A )  ->  ( U. B  C_  A  <->  ( U. B  e.  A  \/  U. B  =  A ) ) )
1513, 14syl6 29 . . . . . . 7  |-  ( Lim 
A  ->  ( B  C_  A  ->  ( U. B  C_  A  <->  ( U. B  e.  A  \/  U. B  =  A ) ) ) )
1615imp 418 . . . . . 6  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( U. B  C_  A  <->  ( U. B  e.  A  \/  U. B  =  A ) ) )
175, 16mpbid 201 . . . . 5  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( U. B  e.  A  \/  U. B  =  A ) )
1817ord 366 . . . 4  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( -.  U. B  e.  A  ->  U. B  =  A ) )
19 cfslb.1 . . . . . . 7  |-  A  e. 
_V
2019cfslb 7908 . . . . . 6  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A )  ~<_  B )
21 domnsym 7003 . . . . . 6  |-  ( ( cf `  A )  ~<_  B  ->  -.  B  ~<  ( cf `  A
) )
2220, 21syl 15 . . . . 5  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  -.  B  ~<  ( cf `  A
) )
23223expia 1153 . . . 4  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( U. B  =  A  ->  -.  B  ~<  ( cf `  A ) ) )
2418, 23syld 40 . . 3  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( -.  U. B  e.  A  ->  -.  B  ~<  ( cf `  A ) ) )
2524con4d 97 . 2  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( B  ~<  ( cf `  A
)  ->  U. B  e.  A ) )
26253impia 1148 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   U.cuni 3843   class class class wbr 4039   Ord word 4407   Oncon0 4408   Lim wlim 4409   ` cfv 5271    ~<_ cdom 6877    ~< csdm 6878   cfccf 7586
This theorem is referenced by:  cfslb2n  7910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-recs 6404  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-card 7588  df-cf 7590
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