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Theorem tz6.26i 25420
 Description: All nonempty (possibly proper) subclasses of , which has a well-founded relation , have -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
tz6.26i.2 Se
Assertion
Ref Expression
tz6.26i
Distinct variable groups:   ,   ,   ,

Proof of Theorem tz6.26i
StepHypRef Expression
1 tz6.26i.1 . 2
2 tz6.26i.2 . 2 Se
3 tz6.26 25419 . 2 Se
41, 2, 3mpanl12 664 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1649   wne 2567  wrex 2667   wss 3280  c0 3588   Se wse 4499   wwe 4500  cpred 25381 This theorem is referenced by:  wfrlem16  25485 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 25382
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