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Theorem onminesb 6522
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 3768 . . 3  |-  ( { x  e.  On  |  ph }  =/=  (/)  <->  E. x  e.  On  ph )
2 ssrab2 3548 . . . 4  |-  { x  e.  On  |  ph }  C_  On
3 onint 6519 . . . 4  |-  ( ( { x  e.  On  |  ph }  C_  On  /\ 
{ x  e.  On  |  ph }  =/=  (/) )  ->  |^| { x  e.  On  |  ph }  e.  {
x  e.  On  |  ph } )
42, 3mpan 670 . . 3  |-  ( { x  e.  On  |  ph }  =/=  (/)  ->  |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph } )
51, 4sylbir 213 . 2  |-  ( E. x  e.  On  ph  ->  |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph } )
6 nfcv 2616 . . . 4  |-  F/_ x On
76elrabsf 3333 . . 3  |-  ( |^| { x  e.  On  |  ph }  e.  { x  e.  On  |  ph }  <->  (
|^| { x  e.  On  |  ph }  e.  On  /\ 
[. |^| { x  e.  On  |  ph }  /  x ]. ph )
)
87simprbi 464 . 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 setvar class
Syntax hints:    -> wi 4    e. wcel 1758    =/= wne 2648   E.wrex 2800   {crab 2803   [.wsbc 3294    C_ wss 3439   (/)c0 3748   |^|cint 4239   Oncon0 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-br 4404  df-opab 4462  df-tr 4497  df-eprel 4743  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834
This theorem is referenced by:  onminex  6531
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