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Theorem tz6.26 23373
 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
StepHypRef Expression
1 wereu2 4283 . . 3 Se
2 reurex 2890 . . 3
31, 2syl 17 . 2 Se
4 rabeq0 3383 . . . 4
5 dfrab3 3351 . . . . . 6
6 vex 2730 . . . . . . 7
76dfpred2 23343 . . . . . 6
85, 7eqtr4i 2276 . . . . 5
98eqeq1i 2260 . . . 4
104, 9bitr3i 244 . . 3
1110rexbii 2532 . 2
123, 11sylib 190 1 Se
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6   wa 360   wceq 1619  cab 2239   wne 2412  wral 2509  wrex 2510  wreu 2511  crab 2512   cin 3077   wss 3078  c0 3362   class class class wbr 3920   Se wse 4243   wwe 4244  cpred 23335 This theorem is referenced by:  tz6.26i  23374  wfi  23375 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-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108 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-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-pred 23336
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