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Theorem pats 30145
 Description: The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
patoms.b
patoms.z
patoms.c
patoms.a
Assertion
Ref Expression
pats
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem pats
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 2966 . 2
2 patoms.a . . 3
3 fveq2 5730 . . . . . 6
4 patoms.b . . . . . 6
53, 4syl6eqr 2488 . . . . 5
6 fveq2 5730 . . . . . . . 8
7 patoms.c . . . . . . . 8
86, 7syl6eqr 2488 . . . . . . 7
98breqd 4225 . . . . . 6
10 fveq2 5730 . . . . . . . 8
11 patoms.z . . . . . . . 8
1210, 11syl6eqr 2488 . . . . . . 7
1312breq1d 4224 . . . . . 6
149, 13bitrd 246 . . . . 5
155, 14rabeqbidv 2953 . . . 4
16 df-ats 30127 . . . 4
17 fvex 5744 . . . . . 6
184, 17eqeltri 2508 . . . . 5
1918rabex 4356 . . . 4
2015, 16, 19fvmpt 5808 . . 3
212, 20syl5eq 2482 . 2
221, 21syl 16 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653   wcel 1726  crab 2711  cvv 2958   class class class wbr 4214  cfv 5456  cbs 13471  cp0 14468   ccvr 30122  catm 30123 This theorem is referenced by:  isat  30146 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-sbc 3164  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-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ats 30127
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