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Theorem onminsb 6631
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 3754 . . 3  |-  ( { x  e.  On  |  ph }  =/=  (/)  <->  E. x  e.  On  ph )
2 ssrab2 3516 . . . 4  |-  { x  e.  On  |  ph }  C_  On
3 onint 6627 . . . 4  |-  ( ( { x  e.  On  |  ph }  C_  On  /\ 
{ x  e.  On  |  ph }  =/=  (/) )  ->  |^| { x  e.  On  |  ph }  e.  {
x  e.  On  |  ph } )
42, 3mpan 677 . . 3  |-  ( { x  e.  On  |  ph }  =/=  (/)  ->  |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph } )
51, 4sylbir 217 . 2  |-  ( E. x  e.  On  ph  ->  |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph } )
6 nfrab1 2973 . . . . 5  |-  F/_ x { x  e.  On  |  ph }
76nfint 4247 . . . 4  |-  F/_ x |^| { x  e.  On  |  ph }
8 nfcv 2594 . . . 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 3196 . . 3  |-  ( |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph }  <->  (
|^| { x  e.  On  |  ph }  e.  On  /\ 
ps ) )
1211simprbi 466 . 2  |-  ( |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph }  ->  ps )
135, 12syl 17 1  |-  ( E. x  e.  On  ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    = wceq 1446   F/wnf 1669    e. wcel 1889    =/= wne 2624   E.wrex 2740   {crab 2743    C_ wss 3406   (/)c0 3733   |^|cint 4237   Oncon0 5426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-br 4406  df-opab 4465  df-tr 4501  df-eprel 4748  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-ord 5429  df-on 5430
This theorem is referenced by:  oawordeulem  7260  rankidb  8276  cardmin2  8437  cardaleph  8525  cardmin  8994
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