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Theorem defge3 25271
 Description: The greatest element of a poset is an element, when it exists, that is greater than the other elements of the poset. Use the idiom when you mean the greatest element of exists. (Contributed by FL, 30-Dec-2011.)
Hypothesis
Ref Expression
defge3.1
Assertion
Ref Expression
defge3
Distinct variable groups:   ,   ,

Proof of Theorem defge3
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 defge3.1 . . . 4
21gepsup 25250 . . 3
3 simpl 443 . . . . . 6
4 dmexg 4939 . . . . . . . 8
51, 4syl5eqel 2367 . . . . . . 7
65adantr 451 . . . . . 6
7 simpr 447 . . . . . 6
81supdef 25262 . . . . . 6
93, 6, 7, 8syl3anc 1182 . . . . 5
109simpld 445 . . . 4
11 eleq1 2343 . . . . . 6
1211anbi2d 684 . . . . 5
13 breq2 4027 . . . . . 6
1413ralbidv 2563 . . . . 5
1512, 14imbi12d 311 . . . 4
1610, 15mpbiri 224 . . 3
172, 16syl 15 . 2
1817anabsi5 790 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1623   wcel 1684  wral 2543  cvv 2788   class class class wbr 4023   cdm 4689  cfv 5255  (class class class)co 5858  cps 14301   cspw 14303  cge 25220 This theorem is referenced by:  defse3  25272 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-undef 6298  df-riota 6304  df-ps 14306  df-spw 14308  df-ge 25248
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