ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  1pru Unicode version

Theorem 1pru 7265
Description: The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
Assertion
Ref Expression
1pru  |-  ( 2nd `  1P )  =  {
x  |  1Q  <Q  x }

Proof of Theorem 1pru
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-i1p 7176 . . 3  |-  1P  =  <. { y  |  y 
<Q  1Q } ,  {
x  |  1Q  <Q  x } >.
21fveq2i 5356 . 2  |-  ( 2nd `  1P )  =  ( 2nd `  <. { y  |  y  <Q  1Q } ,  { x  |  1Q  <Q  x } >. )
3 ltnqex 7258 . . 3  |-  { y  |  y  <Q  1Q }  e.  _V
4 gtnqex 7259 . . 3  |-  { x  |  1Q  <Q  x }  e.  _V
53, 4op2nd 5976 . 2  |-  ( 2nd `  <. { y  |  y  <Q  1Q } ,  { x  |  1Q  <Q  x } >. )  =  { x  |  1Q  <Q  x }
62, 5eqtri 2120 1  |-  ( 2nd `  1P )  =  {
x  |  1Q  <Q  x }
Colors of variables: wff set class
Syntax hints:    = wceq 1299   {cab 2086   <.cop 3477   class class class wbr 3875   ` cfv 5059   2ndc2nd 5968   1Qc1q 6990    <Q cltq 6994   1Pc1p 7001
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-iinf 4440
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-2nd 5970  df-qs 6365  df-ni 7013  df-nqqs 7057  df-ltnqqs 7062  df-i1p 7176
This theorem is referenced by:  1idpru  7300  recexprlem1ssu  7343  recexprlemss1u  7345
  Copyright terms: Public domain W3C validator