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Theorem tz6.26i 24764
Description: All nonempty (possibly proper) subclasses of  A, which has a well-founded relation  R, have  R-minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 14-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
tz6.26i.1  |-  R  We  A
tz6.26i.2  |-  R Se  A
Assertion
Ref Expression
tz6.26i  |-  ( ( B  C_  A  /\  B  =/=  (/) )  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
Distinct variable groups:    y, A    y, B    y, R

Proof of Theorem tz6.26i
StepHypRef Expression
1 tz6.26i.1 . 2  |-  R  We  A
2 tz6.26i.2 . 2  |-  R Se  A
3 tz6.26 24763 . 2  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
41, 2, 3mpanl12 663 1  |-  ( ( B  C_  A  /\  B  =/=  (/) )  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    =/= wne 2521   E.wrex 2620    C_ wss 3228   (/)c0 3531   Se wse 4432    We wwe 4433   Predcpred 24725
This theorem is referenced by:  wfrlem16  24829
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-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
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-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-xp 4777  df-cnv 4779  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-pred 24726
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