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Theorem bnj69 27826
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 27825 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 939    e. wcel 1621    =/= wne 2412   A.wral 2509   E.wrex 2510    C_ wss 3078   (/)c0 3362   class class class wbr 3920    FrSe w-bnj15 27503
This theorem is referenced by:  bnj1228  27827  bnj1523  27887
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-reg 7190  ax-inf2 7226
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-1o 6365  df-bnj17 27498  df-bnj14 27500  df-bnj13 27502  df-bnj15 27504  df-bnj18 27506  df-bnj19 27508
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