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Theorem onminsb 6512
Description: If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses implicit substitution. Theorem Schema 62 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.)
Hypotheses
Ref Expression
onminsb.1  |-  F/ x ps
onminsb.2  |-  ( x  =  |^| { x  e.  On  |  ph }  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
onminsb  |-  ( E. x  e.  On  ph  ->  ps )

Proof of Theorem onminsb
StepHypRef Expression
1 rabn0 3757 . . 3  |-  ( { x  e.  On  |  ph }  =/=  (/)  <->  E. x  e.  On  ph )
2 ssrab2 3537 . . . 4  |-  { x  e.  On  |  ph }  C_  On
3 onint 6508 . . . 4  |-  ( ( { x  e.  On  |  ph }  C_  On  /\ 
{ x  e.  On  |  ph }  =/=  (/) )  ->  |^| { x  e.  On  |  ph }  e.  {
x  e.  On  |  ph } )
42, 3mpan 670 . . 3  |-  ( { x  e.  On  |  ph }  =/=  (/)  ->  |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph } )
51, 4sylbir 213 . 2  |-  ( E. x  e.  On  ph  ->  |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph } )
6 nfrab1 2999 . . . . 5  |-  F/_ x { x  e.  On  |  ph }
76nfint 4238 . . . 4  |-  F/_ x |^| { x  e.  On  |  ph }
8 nfcv 2613 . . . 4  |-  F/_ x On
9 onminsb.1 . . . 4  |-  F/ x ps
10 onminsb.2 . . . 4  |-  ( x  =  |^| { x  e.  On  |  ph }  ->  ( ph  <->  ps )
)
117, 8, 9, 10elrabf 3214 . . 3  |-  ( |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph }  <->  (
|^| { x  e.  On  |  ph }  e.  On  /\ 
ps ) )
1211simprbi 464 . 2  |-  ( |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph }  ->  ps )
135, 12syl 16 1  |-  ( E. x  e.  On  ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370   F/wnf 1590    e. wcel 1758    =/= wne 2644   E.wrex 2796   {crab 2799    C_ wss 3428   (/)c0 3737   |^|cint 4228   Oncon0 4819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-br 4393  df-opab 4451  df-tr 4486  df-eprel 4732  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823
This theorem is referenced by:  oawordeulem  7095  rankidb  8110  cardmin2  8271  cardaleph  8362  cardmin  8831
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