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Theorem dfom5b 24523
Description: A quantifier-free definition of  om that does not depend on ax-inf 7355. (Note: label was changed from dfom5 7367 to dfom5b 24523 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 2804 . . . . . 6  |-  x  e. 
_V
21elint 3884 . . . . 5  |-  ( x  e.  |^| Limits 
<-> 
A. y ( y  e.  Limits  ->  x  e.  y ) )
3 vex 2804 . . . . . . . 8  |-  y  e. 
_V
43ellimits 24521 . . . . . . 7  |-  ( y  e.  Limits 
<->  Lim  y )
54imbi1i 315 . . . . . 6  |-  ( ( y  e.  Limits  ->  x  e.  y )  <->  ( Lim  y  ->  x  e.  y ) )
65albii 1556 . . . . 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 4675 . . 3  |-  ( x  e.  om  <->  ( x  e.  On  /\  A. y
( Lim  y  ->  x  e.  y ) ) )
10 elin 3371 . . 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 2293 1  |-  om  =  ( On  i^i  |^| Limits )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696    i^i cin 3164   |^|cint 3878   Oncon0 4408   Lim wlim 4409   omcom 4672   Limitsclimits 24450
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-int 3879  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-om 4673  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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