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Theorem cfslbn 8432
Description: Any subset of  A smaller than its cofinality has union less than  A. (This is the contrapositive to cfslb 8431.) (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 4109 . . . . . . . 8  |-  ( B 
C_  A  ->  U. B  C_ 
U. A )
2 limuni 4775 . . . . . . . . 9  |-  ( Lim 
A  ->  A  =  U. A )
32sseq2d 3381 . . . . . . . 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 4774 . . . . . . . . . . . 12  |-  ( Lim 
A  ->  Ord  A )
7 ordsson 6400 . . . . . . . . . . . 12  |-  ( Ord 
A  ->  A  C_  On )
86, 7syl 16 . . . . . . . . . . 11  |-  ( Lim 
A  ->  A  C_  On )
9 sstr2 3360 . . . . . . . . . . 11  |-  ( B 
C_  A  ->  ( A  C_  On  ->  B  C_  On ) )
108, 9syl5com 30 . . . . . . . . . 10  |-  ( Lim 
A  ->  ( B  C_  A  ->  B  C_  On ) )
11 ssorduni 6396 . . . . . . . . . 10  |-  ( B 
C_  On  ->  Ord  U. B )
1210, 11syl6 33 . . . . . . . . 9  |-  ( Lim 
A  ->  ( B  C_  A  ->  Ord  U. B
) )
1312, 6jctird 541 . . . . . . . 8  |-  ( Lim 
A  ->  ( B  C_  A  ->  ( Ord  U. B  /\  Ord  A
) ) )
14 ordsseleq 4744 . . . . . . . 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 8431 . . . . . 6  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A )  ~<_  B )
21 domnsym 7433 . . . . . 6  |-  ( ( cf `  A )  ~<_  B  ->  -.  B  ~<  ( cf `  A
) )
2220, 21syl 16 . . . . 5  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  -.  B  ~<  ( cf `  A
) )
23223expia 1184 . . . 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 1179 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 960    = wceq 1364    e. wcel 1761   _Vcvv 2970    C_ wss 3325   U.cuni 4088   class class class wbr 4289   Ord word 4714   Oncon0 4715   Lim wlim 4716   ` cfv 5415    ~<_ cdom 7304    ~< csdm 7305   cfccf 8103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-recs 6828  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-card 8105  df-cf 8107
This theorem is referenced by:  cfslb2n  8433
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