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Theorem dfom5b 25757
Description: A quantifier-free definition of  om that does not depend on ax-inf 7593. (Note: label was changed from dfom5 7605 to dfom5b 25757 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 2959 . . . . . 6  |-  x  e. 
_V
21elint 4056 . . . . 5  |-  ( x  e.  |^| Limits 
<-> 
A. y ( y  e.  Limits  ->  x  e.  y ) )
3 vex 2959 . . . . . . . 8  |-  y  e. 
_V
43ellimits 25755 . . . . . . 7  |-  ( y  e.  Limits 
<->  Lim  y )
54imbi1i 316 . . . . . 6  |-  ( ( y  e.  Limits  ->  x  e.  y )  <->  ( Lim  y  ->  x  e.  y ) )
65albii 1575 . . . . 5  |-  ( A. y ( y  e. 
Limits  ->  x  e.  y )  <->  A. y ( Lim  y  ->  x  e.  y ) )
72, 6bitr2i 242 . . . 4  |-  ( A. y ( Lim  y  ->  x  e.  y )  <-> 
x  e.  |^| Limits )
87anbi2i 676 . . 3  |-  ( ( x  e.  On  /\  A. y ( Lim  y  ->  x  e.  y ) )  <->  ( x  e.  On  /\  x  e. 
|^| Limits ) )
9 elom 4848 . . 3  |-  ( x  e.  om  <->  ( x  e.  On  /\  A. y
( Lim  y  ->  x  e.  y ) ) )
10 elin 3530 . . 3  |-  ( x  e.  ( On  i^i  |^| Limits )  <->  ( x  e.  On  /\  x  e. 
|^| Limits ) )
118, 9, 103bitr4i 269 . 2  |-  ( x  e.  om  <->  x  e.  ( On  i^i  |^| Limits ) )
1211eqriv 2433 1  |-  om  =  ( On  i^i  |^| Limits )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725    i^i cin 3319   |^|cint 4050   Oncon0 4581   Lim wlim 4582   omcom 4845   Limitsclimits 25680
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-1st 6349  df-2nd 6350  df-symdif 25663  df-txp 25698  df-bigcup 25702  df-fix 25703  df-limits 25704
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