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Theorem wereu 4880
Description: A subset of a well-ordered set has a unique minimal element. (Contributed by NM, 18-Mar-1997.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
wereu
Distinct variable groups:   , ,   , ,   , ,

Proof of Theorem wereu
StepHypRef Expression
1 wefr 4874 . . 3
2 fri 4846 . . . . . 6
32exp32 605 . . . . 5
43expcom 435 . . . 4
543imp2 1211 . . 3
61, 5sylan 471 . 2
7 simpr2 1003 . . . 4
8 weso 4875 . . . . 5
98adantr 465 . . . 4
10 soss 4823 . . . 4
117, 9, 10sylc 60 . . 3
12 somo 4839 . . 3
1311, 12syl 16 . 2
14 reu5 3073 . 2
156, 13, 14sylanbrc 664 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  /\wa 369  /\w3a 973  e.wcel 1818  =/=wne 2652  A.wral 2807  E.wrex 2808  E!wreu 2809  E*wrmo 2810  C_wss 3475   c0 3784   class class class wbr 4452  Orwor 4804  Frwfr 4840  Wewwe 4842
This theorem is referenced by:  htalem  8335  zorn2lem1  8897  dyadmax  22007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-po 4805  df-so 4806  df-fr 4843  df-we 4845
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