Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-charfunr Unicode version

Theorem bj-charfunr 13323
 Description: If a class has a "weak" characteristic function on a class , then negated membership in is decidable (in other words, membership in is testable) in . The hypothesis imposes that be a set. As usual, it could be formulated as to deal with general classes, but that extra generality would not make the theorem much more useful. The theorem would still hold if the codomain of were any class with testable equality to the point where is sent. (Contributed by BJ, 6-Aug-2024.)
Hypothesis
Ref Expression
bj-charfunr.1
Assertion
Ref Expression
bj-charfunr DECID
Distinct variable groups:   ,   ,   ,,
Allowed substitution hints:   ()   ()

Proof of Theorem bj-charfunr
StepHypRef Expression
1 bj-charfunr.1 . . . . 5
2 elmapi 6604 . . . . . . . . . 10
3 ffvelrn 5593 . . . . . . . . . . 11
43ex 114 . . . . . . . . . 10
52, 4syl 14 . . . . . . . . 9
6 0elnn 4572 . . . . . . . . . 10
7 nn0eln0 4573 . . . . . . . . . . 11
87orbi2d 780 . . . . . . . . . 10
96, 8mpbid 146 . . . . . . . . 9
105, 9syl6 33 . . . . . . . 8
1110adantr 274 . . . . . . 7
12 elin 3286 . . . . . . . . . . . . . . 15
13 rsp 2501 . . . . . . . . . . . . . . 15
1412, 13syl5bir 152 . . . . . . . . . . . . . 14
1514expd 256 . . . . . . . . . . . . 13
1615adantr 274 . . . . . . . . . . . 12
1716imp 123 . . . . . . . . . . 11
1817necon2bd 2382 . . . . . . . . . 10
19 eldif 3107 . . . . . . . . . . . . . . 15
20 rsp 2501 . . . . . . . . . . . . . . 15
2119, 20syl5bir 152 . . . . . . . . . . . . . 14
2221expd 256 . . . . . . . . . . . . 13
2322adantl 275 . . . . . . . . . . . 12
2423imp 123 . . . . . . . . . . 11
2524necon3ad 2366 . . . . . . . . . 10
2618, 25orim12d 776 . . . . . . . . 9
2726ex 114 . . . . . . . 8
2827adantl 275 . . . . . . 7
2911, 28mpdd 41 . . . . . 6
3029adantl 275 . . . . 5
311, 30rexlimddv 2576 . . . 4
3231imp 123 . . 3
33 df-dc 821 . . 3 DECID
3432, 33sylibr 133 . 2 DECID
3534ralrimiva 2527 1 DECID
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wo 698  DECID wdc 820   wceq 1332   wcel 2125   wne 2324  wral 2432  wrex 2433   cdif 3095   cin 3097  c0 3390  com 4543  wf 5159  cfv 5163  (class class class)co 5814   cmap 6582 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-iinf 4541 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-rex 2438  df-v 2711  df-sbc 2934  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-br 3962  df-opab 4022  df-id 4248  df-suc 4326  df-iom 4544  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-fv 5171  df-ov 5817  df-oprab 5818  df-mpo 5819  df-map 6584 This theorem is referenced by:  bj-charfunbi  13324
 Copyright terms: Public domain W3C validator