MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wereu Structured version   Visualization version   Unicode version

Theorem wereu 4829
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 4823 . . 3  |-  ( R  We  A  ->  R  Fr  A )
2 fri 4795 . . . . . 6  |-  ( ( ( B  e.  V  /\  R  Fr  A
)  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
32exp32 609 . . . . 5  |-  ( ( B  e.  V  /\  R  Fr  A )  ->  ( B  C_  A  ->  ( B  =/=  (/)  ->  E. x  e.  B  A. y  e.  B  -.  y R x ) ) )
43expcom 437 . . . 4  |-  ( R  Fr  A  ->  ( B  e.  V  ->  ( B  C_  A  ->  ( B  =/=  (/)  ->  E. x  e.  B  A. y  e.  B  -.  y R x ) ) ) )
543imp2 1223 . . 3  |-  ( ( R  Fr  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
61, 5sylan 474 . 2  |-  ( ( R  We  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
7 simpr2 1014 . . . 4  |-  ( ( R  We  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  B  C_  A )
8 weso 4824 . . . . 5  |-  ( R  We  A  ->  R  Or  A )
98adantr 467 . . . 4  |-  ( ( R  We  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  R  Or  A )
10 soss 4772 . . . 4  |-  ( B 
C_  A  ->  ( R  Or  A  ->  R  Or  B ) )
117, 9, 10sylc 62 . . 3  |-  ( ( R  We  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  R  Or  B )
12 somo 4788 . . 3  |-  ( R  Or  B  ->  E* x  e.  B  A. y  e.  B  -.  y R x )
1311, 12syl 17 . 2  |-  ( ( R  We  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  E* x  e.  B  A. y  e.  B  -.  y R x )
14 reu5 3007 . 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 669 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 371    /\ w3a 984    e. wcel 1886    =/= wne 2621   A.wral 2736   E.wrex 2737   E!wreu 2738   E*wrmo 2739    C_ wss 3403   (/)c0 3730   class class class wbr 4401    Or wor 4753    Fr wfr 4789    We wwe 4791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-br 4402  df-po 4754  df-so 4755  df-fr 4792  df-we 4794
This theorem is referenced by:  htalem  8364  zorn2lem1  8923  dyadmax  22549  wessf1ornlem  37453
  Copyright terms: Public domain W3C validator