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

Theorem supnub 7423
Description: An upper bound is not less than the supremum. (Contributed by NM, 13-Oct-2004.)
Hypotheses
Ref Expression
supmo.1  |-  ( ph  ->  R  Or  A )
supcl.2  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )
Assertion
Ref Expression
supnub  |-  ( ph  ->  ( ( C  e.  A  /\  A. z  e.  B  -.  C R z )  ->  -.  C R sup ( B ,  A ,  R ) ) )
Distinct variable groups:    x, y,
z, A    x, R, y, z    x, B, y, z    z, C
Allowed substitution hints:    ph( x, y, z)    C( x, y)

Proof of Theorem supnub
StepHypRef Expression
1 supmo.1 . . . . . 6  |-  ( ph  ->  R  Or  A )
2 supcl.2 . . . . . 6  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )
31, 2suplub 7421 . . . . 5  |-  ( ph  ->  ( ( C  e.  A  /\  C R sup ( B ,  A ,  R )
)  ->  E. z  e.  B  C R
z ) )
43expdimp 427 . . . 4  |-  ( (
ph  /\  C  e.  A )  ->  ( C R sup ( B ,  A ,  R
)  ->  E. z  e.  B  C R
z ) )
5 dfrex2 2679 . . . 4  |-  ( E. z  e.  B  C R z  <->  -.  A. z  e.  B  -.  C R z )
64, 5syl6ib 218 . . 3  |-  ( (
ph  /\  C  e.  A )  ->  ( C R sup ( B ,  A ,  R
)  ->  -.  A. z  e.  B  -.  C R z ) )
76con2d 109 . 2  |-  ( (
ph  /\  C  e.  A )  ->  ( A. z  e.  B  -.  C R z  ->  -.  C R sup ( B ,  A ,  R ) ) )
87expimpd 587 1  |-  ( ph  ->  ( ( C  e.  A  /\  A. z  e.  B  -.  C R z )  ->  -.  C R sup ( B ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    e. wcel 1721   A.wral 2666   E.wrex 2667   class class class wbr 4172    Or wor 4462   supcsup 7403
This theorem is referenced by:  supmax  7426  dgrlb  20108  supssd  24051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-po 4463  df-so 4464  df-iota 5377  df-riota 6508  df-sup 7404
  Copyright terms: Public domain W3C validator