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Theorem onminesb 4780
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 3649 . . 3  |-  ( { x  e.  On  |  ph }  =/=  (/)  <->  E. x  e.  On  ph )
2 ssrab2 3430 . . . 4  |-  { x  e.  On  |  ph }  C_  On
3 onint 4777 . . . 4  |-  ( ( { x  e.  On  |  ph }  C_  On  /\ 
{ x  e.  On  |  ph }  =/=  (/) )  ->  |^| { x  e.  On  |  ph }  e.  {
x  e.  On  |  ph } )
42, 3mpan 653 . . 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 2574 . . . 4  |-  F/_ x On
76elrabsf 3201 . . 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 16 1  |-  ( E. x  e.  On  ph  ->  [. |^| { x  e.  On  |  ph }  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726    =/= wne 2601   E.wrex 2708   {crab 2711   [.wsbc 3163    C_ wss 3322   (/)c0 3630   |^|cint 4052   Oncon0 4583
This theorem is referenced by:  onminex  4789
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587
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