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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
supssd.1
supssd.2
supssd.3
supssd.4
Assertion
Ref Expression
supssd
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,
Allowed substitution hints:   (,)

Proof of Theorem supssd
StepHypRef Expression
1 supssd.0 . . . 4
2 supssd.4 . . . 4
31, 2supcl 7225 . . 3
4 supssd.1 . . . . . 6
54sseld 3192 . . . . 5
61, 2supub 7226 . . . . 5
75, 6syld 40 . . . 4
87ralrimiv 2638 . . 3
93, 8jca 518 . 2
10 supssd.3 . . 3
111, 10supnub 7229 . 2
129, 11mpd 14 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 358   wcel 1696  wral 2556  wrex 2557   wss 3165   class class class wbr 4039   wor 4329  csup 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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