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Theorem bnj69 29406
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 238 . 2  |-  ( ( R  FrSe  A  /\  B  C_  A  /\  B  =/=  (/) )  <->  ( R  FrSe  A  /\  B  C_  A  /\  B  =/=  (/) ) )
2 biid 238 . 2  |-  ( ( x  e.  B  /\  y  e.  B  /\  y R x )  <->  ( x  e.  B  /\  y  e.  B  /\  y R x ) )
3 biid 238 . 2  |-  ( A. y  e.  B  -.  y R x  <->  A. y  e.  B  -.  y R x )
41, 2, 3bnj1189 29405 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 976    e. wcel 1844    =/= wne 2600   A.wral 2756   E.wrex 2757    C_ wss 3416   (/)c0 3740   class class class wbr 4397    FrSe w-bnj15 29084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-reg 8054  ax-inf2 8093
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-om 6686  df-1o 7169  df-bnj17 29079  df-bnj14 29081  df-bnj13 29083  df-bnj15 29085  df-bnj18 29087  df-bnj19 29089
This theorem is referenced by:  bnj1228  29407  bnj1523  29467
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