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Theorem dfom5b 24452
Description: A quantifier-free definition of  om that does not depend on ax-inf 7339. (Note: label was changed from dfom5 7351 to dfom5b 24452 to prevent naming conflict. NM 12-Feb-2013) (Contributed by Scott Fenton, 11-Apr-2012.)
Assertion
Ref Expression
dfom5b  |-  om  =  ( On  i^i  |^| Limits )

Proof of Theorem dfom5b
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . . . 6  |-  x  e. 
_V
21elint 3868 . . . . 5  |-  ( x  e.  |^| Limits 
<-> 
A. y ( y  e.  Limits  ->  x  e.  y ) )
3 vex 2791 . . . . . . . 8  |-  y  e. 
_V
43ellimits 24450 . . . . . . 7  |-  ( y  e.  Limits 
<->  Lim  y )
54imbi1i 315 . . . . . 6  |-  ( ( y  e.  Limits  ->  x  e.  y )  <->  ( Lim  y  ->  x  e.  y ) )
65albii 1553 . . . . 5  |-  ( A. y ( y  e. 
Limits  ->  x  e.  y )  <->  A. y ( Lim  y  ->  x  e.  y ) )
72, 6bitr2i 241 . . . 4  |-  ( A. y ( Lim  y  ->  x  e.  y )  <-> 
x  e.  |^| Limits )
87anbi2i 675 . . 3  |-  ( ( x  e.  On  /\  A. y ( Lim  y  ->  x  e.  y ) )  <->  ( x  e.  On  /\  x  e. 
|^| Limits ) )
9 elom 4659 . . 3  |-  ( x  e.  om  <->  ( x  e.  On  /\  A. y
( Lim  y  ->  x  e.  y ) ) )
10 elin 3358 . . 3  |-  ( x  e.  ( On  i^i  |^| Limits )  <->  ( x  e.  On  /\  x  e. 
|^| Limits ) )
118, 9, 103bitr4i 268 . 2  |-  ( x  e.  om  <->  x  e.  ( On  i^i  |^| Limits ) )
1211eqriv 2280 1  |-  om  =  ( On  i^i  |^| Limits )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684    i^i cin 3151   |^|cint 3862   Oncon0 4392   Lim wlim 4393   omcom 4656   Limitsclimits 24379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-symdif 24362  df-txp 24395  df-bigcup 24399  df-fix 24400  df-limits 24401
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