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Theorem wereu 4715
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 4709 . . 3  |-  ( R  We  A  ->  R  Fr  A )
2 fri 4681 . . . . . 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 1202 . . 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 995 . . . 4  |-  ( ( R  We  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  B  C_  A )
8 weso 4710 . . . . 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 4658 . . . 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 4674 . . 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 2935 . 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 965    e. wcel 1756    =/= wne 2605   A.wral 2714   E.wrex 2715   E!wreu 2716   E*wrmo 2717    C_ wss 3327   (/)c0 3636   class class class wbr 4291    Or wor 4639    Fr wfr 4675    We wwe 4677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-br 4292  df-po 4640  df-so 4641  df-fr 4678  df-we 4680
This theorem is referenced by:  htalem  8102  zorn2lem1  8664  dyadmax  21077
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