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Theorem wereu 4864
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 4858 . . 3  |-  ( R  We  A  ->  R  Fr  A )
2 fri 4830 . . . . . 6  |-  ( ( ( B  e.  V  /\  R  Fr  A
)  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
32exp32 603 . . . . 5  |-  ( ( B  e.  V  /\  R  Fr  A )  ->  ( B  C_  A  ->  ( B  =/=  (/)  ->  E. x  e.  B  A. y  e.  B  -.  y R x ) ) )
43expcom 433 . . . 4  |-  ( R  Fr  A  ->  ( B  e.  V  ->  ( B  C_  A  ->  ( B  =/=  (/)  ->  E. x  e.  B  A. y  e.  B  -.  y R x ) ) ) )
543imp2 1209 . . 3  |-  ( ( R  Fr  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
61, 5sylan 469 . 2  |-  ( ( R  We  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
7 simpr2 1001 . . . 4  |-  ( ( R  We  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  B  C_  A )
8 weso 4859 . . . . 5  |-  ( R  We  A  ->  R  Or  A )
98adantr 463 . . . 4  |-  ( ( R  We  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  R  Or  A )
10 soss 4807 . . . 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 4823 . . 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 3070 . 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 662 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 367    /\ w3a 971    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   E!wreu 2806   E*wrmo 2807    C_ wss 3461   (/)c0 3783   class class class wbr 4439    Or wor 4788    Fr wfr 4824    We wwe 4826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-po 4789  df-so 4790  df-fr 4827  df-we 4829
This theorem is referenced by:  htalem  8305  zorn2lem1  8867  dyadmax  22173
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