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Theorem onminsb 4693
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 3562 . . 3  |-  ( { x  e.  On  |  ph }  =/=  (/)  <->  E. x  e.  On  ph )
2 ssrab2 3344 . . . 4  |-  { x  e.  On  |  ph }  C_  On
3 onint 4689 . . . 4  |-  ( ( { x  e.  On  |  ph }  C_  On  /\ 
{ x  e.  On  |  ph }  =/=  (/) )  ->  |^| { x  e.  On  |  ph }  e.  {
x  e.  On  |  ph } )
42, 3mpan 651 . . 3  |-  ( { x  e.  On  |  ph }  =/=  (/)  ->  |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph } )
51, 4sylbir 204 . 2  |-  ( E. x  e.  On  ph  ->  |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph } )
6 nfrab1 2805 . . . . 5  |-  F/_ x { x  e.  On  |  ph }
76nfint 3974 . . . 4  |-  F/_ x |^| { x  e.  On  |  ph }
8 nfcv 2502 . . . 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 3008 . . 3  |-  ( |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph }  <->  (
|^| { x  e.  On  |  ph }  e.  On  /\ 
ps ) )
1211simprbi 450 . 2  |-  ( |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph }  ->  ps )
135, 12syl 15 1  |-  ( E. x  e.  On  ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   F/wnf 1549    = wceq 1647    e. wcel 1715    =/= wne 2529   E.wrex 2629   {crab 2632    C_ wss 3238   (/)c0 3543   |^|cint 3964   Oncon0 4495
This theorem is referenced by:  oawordeulem  6694  rankidb  7619  cardmin2  7778  cardaleph  7863  cardmin  8333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-br 4126  df-opab 4180  df-tr 4216  df-eprel 4408  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499
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