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Theorem fimax2gOLD 26017
Description: A finite set has a maximum under a total order. (Moved to fimax2g 7250 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Mar-2013.) (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
fimax2gOLD  |-  ( ( R  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  -.  x R y )
Distinct variable groups:    x, R, y    x, A, y

Proof of Theorem fimax2gOLD
StepHypRef Expression
1 fimax2g 7250 1  |-  ( ( R  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  -.  x R y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 935    e. wcel 1715    =/= wne 2529   A.wral 2628   E.wrex 2629   (/)c0 3543   class class class wbr 4125    Or wor 4416   Fincfn 7006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-1o 6621  df-er 6802  df-en 7007  df-fin 7010
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