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Theorem wereu 4875
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  |-  ( ( R  We  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  E! x  e.  B  A. y  e.  B  -.  y R x )
Distinct variable groups:    x, y, A    x, B, y    x, R, y
Allowed substitution hints:    V( x, y)

Proof of Theorem wereu
StepHypRef Expression
1 wefr 4869 . . 3  |-  ( R  We  A  ->  R  Fr  A )
2 fri 4841 . . . . . 6  |-  ( ( ( B  e.  V  /\  R  Fr  A
)  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
32exp32 605 . . . . 5  |-  ( ( B  e.  V  /\  R  Fr  A )  ->  ( B  C_  A  ->  ( B  =/=  (/)  ->  E. x  e.  B  A. y  e.  B  -.  y R x ) ) )
43expcom 435 . . . 4  |-  ( R  Fr  A  ->  ( B  e.  V  ->  ( B  C_  A  ->  ( B  =/=  (/)  ->  E. x  e.  B  A. y  e.  B  -.  y R x ) ) ) )
543imp2 1211 . . 3  |-  ( ( R  Fr  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
61, 5sylan 471 . 2  |-  ( ( R  We  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
7 simpr2 1003 . . . 4  |-  ( ( R  We  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  B  C_  A )
8 weso 4870 . . . . 5  |-  ( R  We  A  ->  R  Or  A )
98adantr 465 . . . 4  |-  ( ( R  We  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  R  Or  A )
10 soss 4818 . . . 4  |-  ( B 
C_  A  ->  ( R  Or  A  ->  R  Or  B ) )
117, 9, 10sylc 60 . . 3  |-  ( ( R  We  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  R  Or  B )
12 somo 4834 . . 3  |-  ( R  Or  B  ->  E* x  e.  B  A. y  e.  B  -.  y R x )
1311, 12syl 16 . 2  |-  ( ( R  We  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  E* x  e.  B  A. y  e.  B  -.  y R x )
14 reu5 3077 . 2  |-  ( E! x  e.  B  A. y  e.  B  -.  y R x  <->  ( E. x  e.  B  A. y  e.  B  -.  y R x  /\  E* x  e.  B  A. y  e.  B  -.  y R x ) )
156, 13, 14sylanbrc 664 1  |-  ( ( R  We  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  E! x  e.  B  A. y  e.  B  -.  y R x )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   E!wreu 2816   E*wrmo 2817    C_ wss 3476   (/)c0 3785   class class class wbr 4447    Or wor 4799    Fr wfr 4835    We wwe 4837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-po 4800  df-so 4801  df-fr 4838  df-we 4840
This theorem is referenced by:  htalem  8314  zorn2lem1  8876  dyadmax  21770
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