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Theorem tz6.26 25480
 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, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
tz6.26 Se
Distinct variable groups:   ,   ,   ,

Proof of Theorem tz6.26
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 wereu2 4579 . . 3 Se
2 reurex 2922 . . 3
31, 2syl 16 . 2 Se
4 rabeq0 3649 . . . 4
5 dfrab3 3617 . . . . . 6
6 vex 2959 . . . . . . 7
76dfpred2 25448 . . . . . 6
85, 7eqtr4i 2459 . . . . 5
98eqeq1i 2443 . . . 4
104, 9bitr3i 243 . . 3
1110rexbii 2730 . 2
123, 11sylib 189 1 Se
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359   wceq 1652  cab 2422   wne 2599  wral 2705  wrex 2706  wreu 2707  crab 2709   cin 3319   wss 3320  c0 3628   class class class wbr 4212   Se wse 4539   wwe 4540  cpred 25438 This theorem is referenced by:  tz6.26i  25481  wfi  25482  wzel  25575  wsuclem  25576 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-pred 25439
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