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Theorem bnj69 28307
Description: Existence of a minimal element in certain classes: if  R is well-founded and set-like on 
A, then every non-empty subclass of  A has a minimal element. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj69  |-  ( ( R  FrSe  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
Distinct variable groups:    x, A, y    x, B, y    x, R, y

Proof of Theorem bnj69
StepHypRef Expression
1 biid 229 . 2  |-  ( ( R  FrSe  A  /\  B  C_  A  /\  B  =/=  (/) )  <->  ( R  FrSe  A  /\  B  C_  A  /\  B  =/=  (/) ) )
2 biid 229 . 2  |-  ( ( x  e.  B  /\  y  e.  B  /\  y R x )  <->  ( x  e.  B  /\  y  e.  B  /\  y R x ) )
3 biid 229 . 2  |-  ( A. y  e.  B  -.  y R x  <->  A. y  e.  B  -.  y R x )
41, 2, 3bnj1189 28306 1  |-  ( ( R  FrSe  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ w3a 936    e. wcel 1685    =/= wne 2447   A.wral 2544   E.wrex 2545    C_ wss 3153   (/)c0 3456   class class class wbr 4024    FrSe w-bnj15 27984
This theorem is referenced by:  bnj1228  28308  bnj1523  28368
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-reg 7301  ax-inf2 7337
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  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-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-1o 6474  df-bnj17 27979  df-bnj14 27981  df-bnj13 27983  df-bnj15 27985  df-bnj18 27987  df-bnj19 27989
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