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Theorem bnj69 32356
Description: Existence of a minimal element in certain classes: if  R is well-founded and set-like on 
A, then every nonempty 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 236 . 2  |-  ( ( R  FrSe  A  /\  B  C_  A  /\  B  =/=  (/) )  <->  ( R  FrSe  A  /\  B  C_  A  /\  B  =/=  (/) ) )
2 biid 236 . 2  |-  ( ( x  e.  B  /\  y  e.  B  /\  y R x )  <->  ( x  e.  B  /\  y  e.  B  /\  y R x ) )
3 biid 236 . 2  |-  ( A. y  e.  B  -.  y R x  <->  A. y  e.  B  -.  y R x )
41, 2, 3bnj1189 32355 1  |-  ( ( R  FrSe  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 965    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800    C_ wss 3439   (/)c0 3748   class class class wbr 4403    FrSe w-bnj15 32035
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-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-reg 7922  ax-inf2 7962
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  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-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-om 6590  df-1o 7033  df-bnj17 32030  df-bnj14 32032  df-bnj13 32034  df-bnj15 32036  df-bnj18 32038  df-bnj19 32040
This theorem is referenced by:  bnj1228  32357  bnj1523  32417
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