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Theorem onssmin 4670
Description: A non-empty class of ordinal numbers has the smallest member. Exercise 9 of [TakeutiZaring] p. 40. (Contributed by NM, 3-Oct-2003.)
Assertion
Ref Expression
onssmin  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  x  C_  y
)
Distinct variable group:    x, y, A

Proof of Theorem onssmin
StepHypRef Expression
1 onint 4668 . 2  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )
2 intss1 3958 . . 3  |-  ( y  e.  A  ->  |^| A  C_  y )
32rgen 2684 . 2  |-  A. y  e.  A  |^| A  C_  y
4 sseq1 3275 . . . 4  |-  ( x  =  |^| A  -> 
( x  C_  y  <->  |^| A  C_  y )
)
54ralbidv 2639 . . 3  |-  ( x  =  |^| A  -> 
( A. y  e.  A  x  C_  y  <->  A. y  e.  A  |^| A  C_  y ) )
65rspcev 2960 . 2  |-  ( (
|^| A  e.  A  /\  A. y  e.  A  |^| A  C_  y )  ->  E. x  e.  A  A. y  e.  A  x  C_  y )
71, 3, 6sylancl 643 1  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  x  C_  y
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   E.wrex 2620    C_ wss 3228   (/)c0 3531   |^|cint 3943   Oncon0 4474
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-br 4105  df-opab 4159  df-tr 4195  df-eprel 4387  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478
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