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Theorem onminesb 4588
Description: If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses explicit substitution. Theorem Schema 62 of [Suppes] p. 228. (Contributed by NM, 29-Sep-2003.)
Assertion
Ref Expression
onminesb  |-  ( E. x  e.  On  ph  ->  [. |^| { x  e.  On  |  ph }  /  x ]. ph )

Proof of Theorem onminesb
StepHypRef Expression
1 rabn0 3475 . . 3  |-  ( { x  e.  On  |  ph }  =/=  (/)  <->  E. x  e.  On  ph )
2 ssrab2 3259 . . . 4  |-  { x  e.  On  |  ph }  C_  On
3 onint 4585 . . . 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 nfcv 2420 . . . 4  |-  F/_ x On
76elrabsf 3030 . . 3  |-  ( |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph }  <->  (
|^| { x  e.  On  |  ph }  e.  On  /\ 
[. |^| { x  e.  On  |  ph }  /  x ]. ph )
)
87simprbi 452 . 2  |-  ( |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph }  ->  [. |^| { x  e.  On  |  ph }  /  x ]. ph )
95, 8syl 17 1  |-  ( E. x  e.  On  ph  ->  [. |^| { x  e.  On  |  ph }  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1688    =/= wne 2447   E.wrex 2545   {crab 2548   [.wsbc 2992    C_ wss 3153   (/)c0 3456   |^|cint 3863   Oncon0 4391
This theorem is referenced by:  onminex  4597
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
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 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-br 4025  df-opab 4079  df-tr 4115  df-eprel 4304  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395
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