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Theorem onminsb 4562
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 3449 . . 3  |-  ( { x  e.  On  |  ph }  =/=  (/)  <->  E. x  e.  On  ph )
2 ssrab2 3233 . . . 4  |-  { x  e.  On  |  ph }  C_  On
3 onint 4558 . . . 4  |-  ( ( { x  e.  On  |  ph }  C_  On  /\ 
{ x  e.  On  |  ph }  =/=  (/) )  ->  |^| { x  e.  On  |  ph }  e.  {
x  e.  On  |  ph } )
42, 3mpan 654 . . 3  |-  ( { x  e.  On  |  ph }  =/=  (/)  ->  |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph } )
51, 4sylbir 206 . 2  |-  ( E. x  e.  On  ph  ->  |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph } )
6 nfrab1 2695 . . . . 5  |-  F/_ x { x  e.  On  |  ph }
76nfint 3846 . . . 4  |-  F/_ x |^| { x  e.  On  |  ph }
8 nfcv 2394 . . . 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 2897 . . 3  |-  ( |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph }  <->  (
|^| { x  e.  On  |  ph }  e.  On  /\ 
ps ) )
1211simprbi 452 . 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 set class
Syntax hints:    -> wi 6    <-> wb 178   F/wnf 1539    = wceq 1619    e. wcel 1621    =/= wne 2421   E.wrex 2519   {crab 2522    C_ wss 3127   (/)c0 3430   |^|cint 3836   Oncon0 4364
This theorem is referenced by:  oawordeulem  6520  rankidb  7440  cardmin2  7599  cardaleph  7684  cardmin  8154
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-br 3998  df-opab 4052  df-tr 4088  df-eprel 4277  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368
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