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Theorem dirge 14684
 Description: For any two elements of a directed set, there exists a third element greater than or equal to both. (Note that this does not say that the two elements have a least upper bound.) (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
Hypothesis
Ref Expression
dirge.1
Assertion
Ref Expression
dirge
Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem dirge
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dirge.1 . . . . . . 7
2 dirdm 14681 . . . . . . 7
31, 2syl5eq 2482 . . . . . 6
43eleq2d 2505 . . . . 5
53eleq2d 2505 . . . . 5
64, 5anbi12d 693 . . . 4
7 eqid 2438 . . . . . . . . . 10
87isdir 14679 . . . . . . . . 9
98ibi 234 . . . . . . . 8
109simprd 451 . . . . . . 7
1110simprd 451 . . . . . 6
12 codir 5256 . . . . . 6
1311, 12sylib 190 . . . . 5
14 breq1 4217 . . . . . . . 8
1514anbi1d 687 . . . . . . 7
1615exbidv 1637 . . . . . 6
17 breq1 4217 . . . . . . . 8
1817anbi2d 686 . . . . . . 7
1918exbidv 1637 . . . . . 6
2016, 19rspc2v 3060 . . . . 5
2113, 20syl5com 29 . . . 4
226, 21sylbid 208 . . 3
23 reldir 14680 . . . . . . . . . 10
24 relelrn 5105 . . . . . . . . . 10
2523, 24sylan 459 . . . . . . . . 9
2625ex 425 . . . . . . . 8
27 ssun2 3513 . . . . . . . . . . 11
28 dmrnssfld 5131 . . . . . . . . . . 11
2927, 28sstri 3359 . . . . . . . . . 10
3029, 3syl5sseqr 3399 . . . . . . . . 9
3130sseld 3349 . . . . . . . 8
3226, 31syld 43 . . . . . . 7
3332adantrd 456 . . . . . 6
3433ancrd 539 . . . . 5
3534eximdv 1633 . . . 4
36 df-rex 2713 . . . 4
3735, 36syl6ibr 220 . . 3
3822, 37syld 43 . 2
39383impib 1152 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937  wex 1551   wceq 1653   wcel 1726  wral 2707  wrex 2708   cun 3320   wss 3322  cuni 4017   class class class wbr 4214   cid 4495   cxp 4878  ccnv 4879   cdm 4880   crn 4881   cres 4882   ccom 4884   wrel 4885  cdir 14675 This theorem is referenced by:  tailfb  26408 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-dir 14677
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