MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cfslbn Unicode version

Theorem cfslbn 8073
Description: Any subset of  A smaller than its cofinality has union less than  A. (This is the contrapositive to cfslb 8072.) (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 3971 . . . . . . . 8  |-  ( B 
C_  A  ->  U. B  C_ 
U. A )
2 limuni 4575 . . . . . . . . 9  |-  ( Lim 
A  ->  A  =  U. A )
32sseq2d 3312 . . . . . . . 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 4574 . . . . . . . . . . . 12  |-  ( Lim 
A  ->  Ord  A )
7 ordsson 4703 . . . . . . . . . . . 12  |-  ( Ord 
A  ->  A  C_  On )
86, 7syl 16 . . . . . . . . . . 11  |-  ( Lim 
A  ->  A  C_  On )
9 sstr2 3291 . . . . . . . . . . 11  |-  ( B 
C_  A  ->  ( A  C_  On  ->  B  C_  On ) )
108, 9syl5com 28 . . . . . . . . . 10  |-  ( Lim 
A  ->  ( B  C_  A  ->  B  C_  On ) )
11 ssorduni 4699 . . . . . . . . . 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 4544 . . . . . . . 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 8072 . . . . . 6  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A )  ~<_  B )
21 domnsym 7162 . . . . . 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 1649    e. wcel 1717   _Vcvv 2892    C_ wss 3256   U.cuni 3950   class class class wbr 4146   Ord word 4514   Oncon0 4515   Lim wlim 4516   ` cfv 5387    ~<_ cdom 7036    ~< csdm 7037   cfccf 7750
This theorem is referenced by:  cfslb2n  8074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-riota 6478  df-recs 6562  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-card 7752  df-cf 7754
  Copyright terms: Public domain W3C validator