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Theorem ellimits 24521
Description: Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
Hypothesis
Ref Expression
ellimits.1  |-  A  e. 
_V
Assertion
Ref Expression
ellimits  |-  ( A  e.  Limits 
<->  Lim  A )

Proof of Theorem ellimits
StepHypRef Expression
1 df-limits 24472 . . 3  |-  Limits  =  ( ( On  i^i  Fix Bigcup )  \  { (/) } )
21eleq2i 2360 . 2  |-  ( A  e.  Limits 
<->  A  e.  ( ( On  i^i  Fix Bigcup ) 
\  { (/) } ) )
3 eldif 3175 . 2  |-  ( A  e.  ( ( On 
i^i  Fix Bigcup )  \  { (/)
} )  <->  ( A  e.  ( On  i^i  Fix Bigcup )  /\  -.  A  e. 
{ (/) } ) )
4 3anan32 946 . . 3  |-  ( ( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  <->  ( ( Ord  A  /\  A  = 
U. A )  /\  A  =/=  (/) ) )
5 df-lim 4413 . . 3  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
6 elin 3371 . . . . 5  |-  ( A  e.  ( On  i^i  Fix Bigcup )  <->  ( A  e.  On  /\  A  e. 
Fix Bigcup ) )
7 ellimits.1 . . . . . . 7  |-  A  e. 
_V
87elon 4417 . . . . . 6  |-  ( A  e.  On  <->  Ord  A )
97elfix 24514 . . . . . . 7  |-  ( A  e.  Fix Bigcup  <->  A Bigcup A )
107brbigcup 24509 . . . . . . 7  |-  ( A
Bigcup A  <->  U. A  =  A )
11 eqcom 2298 . . . . . . 7  |-  ( U. A  =  A  <->  A  =  U. A )
129, 10, 113bitri 262 . . . . . 6  |-  ( A  e.  Fix Bigcup  <->  A  =  U. A )
138, 12anbi12i 678 . . . . 5  |-  ( ( A  e.  On  /\  A  e.  Fix Bigcup )  <->  ( Ord  A  /\  A  =  U. A ) )
146, 13bitri 240 . . . 4  |-  ( A  e.  ( On  i^i  Fix Bigcup )  <->  ( Ord  A  /\  A  =  U. A ) )
157elsnc 3676 . . . . 5  |-  ( A  e.  { (/) }  <->  A  =  (/) )
1615necon3bbii 2490 . . . 4  |-  ( -.  A  e.  { (/) }  <-> 
A  =/=  (/) )
1714, 16anbi12i 678 . . 3  |-  ( ( A  e.  ( On 
i^i  Fix Bigcup )  /\  -.  A  e.  { (/) } )  <-> 
( ( Ord  A  /\  A  =  U. A )  /\  A  =/=  (/) ) )
184, 5, 173bitr4ri 269 . 2  |-  ( ( A  e.  ( On 
i^i  Fix Bigcup )  /\  -.  A  e.  { (/) } )  <->  Lim  A )
192, 3, 183bitri 262 1  |-  ( A  e.  Limits 
<->  Lim  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    \ cdif 3162    i^i cin 3164   (/)c0 3468   {csn 3653   U.cuni 3843   class class class wbr 4039   Ord word 4407   Oncon0 4408   Lim wlim 4409   Bigcupcbigcup 24448   Fixcfix 24449   Limitsclimits 24450
This theorem is referenced by:  limitssson  24522  dfom5b  24523  dfrdg4  24560
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-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-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-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  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-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139  df-symdif 24433  df-txp 24466  df-bigcup 24470  df-fix 24471  df-limits 24472
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