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Theorem onminsb 4738
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 3607 . . 3  |-  ( { x  e.  On  |  ph }  =/=  (/)  <->  E. x  e.  On  ph )
2 ssrab2 3388 . . . 4  |-  { x  e.  On  |  ph }  C_  On
3 onint 4734 . . . 4  |-  ( ( { x  e.  On  |  ph }  C_  On  /\ 
{ x  e.  On  |  ph }  =/=  (/) )  ->  |^| { x  e.  On  |  ph }  e.  {
x  e.  On  |  ph } )
42, 3mpan 652 . . 3  |-  ( { x  e.  On  |  ph }  =/=  (/)  ->  |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph } )
51, 4sylbir 205 . 2  |-  ( E. x  e.  On  ph  ->  |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph } )
6 nfrab1 2848 . . . . 5  |-  F/_ x { x  e.  On  |  ph }
76nfint 4020 . . . 4  |-  F/_ x |^| { x  e.  On  |  ph }
8 nfcv 2540 . . . 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 3051 . . 3  |-  ( |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph }  <->  (
|^| { x  e.  On  |  ph }  e.  On  /\ 
ps ) )
1211simprbi 451 . 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 set class
Syntax hints:    -> wi 4    <-> wb 177   F/wnf 1550    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   {crab 2670    C_ wss 3280   (/)c0 3588   |^|cint 4010   Oncon0 4541
This theorem is referenced by:  oawordeulem  6756  rankidb  7682  cardmin2  7841  cardaleph  7926  cardmin  8395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545
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