Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nfvres | Unicode version |
Description: The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.) |
Ref | Expression |
---|---|
nfvres |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fv 4930 | . . . . . . . . . 10 | |
2 | df-iota 4887 | . . . . . . . . . 10 | |
3 | 1, 2 | eqtri 2101 | . . . . . . . . 9 |
4 | 3 | eleq2i 2145 | . . . . . . . 8 |
5 | eluni 3604 | . . . . . . . 8 | |
6 | 4, 5 | bitri 182 | . . . . . . 7 |
7 | exsimpr 1549 | . . . . . . 7 | |
8 | 6, 7 | sylbi 119 | . . . . . 6 |
9 | df-clab 2068 | . . . . . . . 8 | |
10 | nfv 1461 | . . . . . . . . 9 | |
11 | sneq 3409 | . . . . . . . . . 10 | |
12 | 11 | eqeq2d 2092 | . . . . . . . . 9 |
13 | 10, 12 | sbie 1714 | . . . . . . . 8 |
14 | 9, 13 | bitri 182 | . . . . . . 7 |
15 | 14 | exbii 1536 | . . . . . 6 |
16 | 8, 15 | sylib 120 | . . . . 5 |
17 | euabsn2 3461 | . . . . 5 | |
18 | 16, 17 | sylibr 132 | . . . 4 |
19 | euex 1971 | . . . 4 | |
20 | df-br 3786 | . . . . . . . 8 | |
21 | df-res 4375 | . . . . . . . . 9 | |
22 | 21 | eleq2i 2145 | . . . . . . . 8 |
23 | 20, 22 | bitri 182 | . . . . . . 7 |
24 | elin 3155 | . . . . . . . 8 | |
25 | 24 | simprbi 269 | . . . . . . 7 |
26 | 23, 25 | sylbi 119 | . . . . . 6 |
27 | opelxp1 4395 | . . . . . 6 | |
28 | 26, 27 | syl 14 | . . . . 5 |
29 | 28 | exlimiv 1529 | . . . 4 |
30 | 18, 19, 29 | 3syl 17 | . . 3 |
31 | 30 | con3i 594 | . 2 |
32 | 31 | eq0rdv 3288 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wceq 1284 wex 1421 wcel 1433 wsb 1685 weu 1941 cab 2067 cvv 2601 cin 2972 c0 3251 csn 3398 cop 3401 cuni 3601 class class class wbr 3785 cxp 4361 cres 4365 cio 4885 cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-res 4375 df-iota 4887 df-fv 4930 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |