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Theorem supssd 23263
Description: Inequality deduction for supremum of a subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)
Hypotheses
Ref Expression
supssd.0  |-  ( ph  ->  R  Or  A )
supssd.1  |-  ( ph  ->  B  C_  C )
supssd.2  |-  ( ph  ->  C  C_  A )
supssd.3  |-  ( 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 ) ) )
supssd.4  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  C  y R z ) ) )
Assertion
Ref Expression
supssd  |-  ( ph  ->  -.  sup ( C ,  A ,  R
) R sup ( B ,  A ,  R ) )
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, C, y, z    x, R, y, z    ph, z
Allowed substitution hints:    ph( x, y)

Proof of Theorem supssd
StepHypRef Expression
1 supssd.0 . . . 4  |-  ( ph  ->  R  Or  A )
2 supssd.4 . . . 4  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  C  y R z ) ) )
31, 2supcl 7225 . . 3  |-  ( ph  ->  sup ( C ,  A ,  R )  e.  A )
4 supssd.1 . . . . . 6  |-  ( ph  ->  B  C_  C )
54sseld 3192 . . . . 5  |-  ( ph  ->  ( z  e.  B  ->  z  e.  C ) )
61, 2supub 7226 . . . . 5  |-  ( ph  ->  ( z  e.  C  ->  -.  sup ( C ,  A ,  R
) R z ) )
75, 6syld 40 . . . 4  |-  ( ph  ->  ( z  e.  B  ->  -.  sup ( C ,  A ,  R
) R z ) )
87ralrimiv 2638 . . 3  |-  ( ph  ->  A. z  e.  B  -.  sup ( C ,  A ,  R ) R z )
93, 8jca 518 . 2  |-  ( ph  ->  ( sup ( C ,  A ,  R
)  e.  A  /\  A. z  e.  B  -.  sup ( C ,  A ,  R ) R z ) )
10 supssd.3 . . 3  |-  ( 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 ) ) )
111, 10supnub 7229 . 2  |-  ( ph  ->  ( ( sup ( C ,  A ,  R )  e.  A  /\  A. z  e.  B  -.  sup ( C ,  A ,  R ) R z )  ->  -.  sup ( C ,  A ,  R ) R sup ( B ,  A ,  R )
) )
129, 11mpd 14 1  |-  ( ph  ->  -.  sup ( C ,  A ,  R
) R sup ( B ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   class class class wbr 4039    Or wor 4329   supcsup 7209
This theorem is referenced by:  xrsupssd  23269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-po 4330  df-so 4331  df-iota 5235  df-riota 6320  df-sup 7210
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