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Theorem p0val 14472
 Description: Value of poset zero. (Contributed by NM, 12-Oct-2011.)
Hypotheses
Ref Expression
p0val.b
p0val.g
p0val.z
Assertion
Ref Expression
p0val

Proof of Theorem p0val
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 2966 . 2
2 p0val.z . . 3
3 fveq2 5730 . . . . . 6
4 p0val.g . . . . . 6
53, 4syl6eqr 2488 . . . . 5
6 fveq2 5730 . . . . . 6
7 p0val.b . . . . . 6
86, 7syl6eqr 2488 . . . . 5
95, 8fveq12d 5736 . . . 4
10 df-p0 14470 . . . 4
11 fvex 5744 . . . 4
129, 10, 11fvmpt 5808 . . 3
132, 12syl5eq 2482 . 2
141, 13syl 16 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653   wcel 1726  cvv 2958  cfv 5456  cbs 13471  cglb 14402  cp0 14468 This theorem is referenced by:  p0le  14474  clatp0ex  24195  xrsp0  24205 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-p0 14470
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