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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 4910 | . . . . . . . . . 10 | |
2 | df-iota 4867 | . . . . . . . . . 10 | |
3 | 1, 2 | eqtri 2060 | . . . . . . . . 9 |
4 | 3 | eleq2i 2104 | . . . . . . . 8 |
5 | eluni 3583 | . . . . . . . 8 | |
6 | 4, 5 | bitri 173 | . . . . . . 7 |
7 | exsimpr 1509 | . . . . . . 7 | |
8 | 6, 7 | sylbi 114 | . . . . . 6 |
9 | df-clab 2027 | . . . . . . . 8 | |
10 | nfv 1421 | . . . . . . . . 9 | |
11 | sneq 3386 | . . . . . . . . . 10 | |
12 | 11 | eqeq2d 2051 | . . . . . . . . 9 |
13 | 10, 12 | sbie 1674 | . . . . . . . 8 |
14 | 9, 13 | bitri 173 | . . . . . . 7 |
15 | 14 | exbii 1496 | . . . . . 6 |
16 | 8, 15 | sylib 127 | . . . . 5 |
17 | euabsn2 3439 | . . . . 5 | |
18 | 16, 17 | sylibr 137 | . . . 4 |
19 | euex 1930 | . . . 4 | |
20 | df-br 3765 | . . . . . . . 8 | |
21 | df-res 4357 | . . . . . . . . 9 | |
22 | 21 | eleq2i 2104 | . . . . . . . 8 |
23 | 20, 22 | bitri 173 | . . . . . . 7 |
24 | elin 3126 | . . . . . . . 8 | |
25 | 24 | simprbi 260 | . . . . . . 7 |
26 | 23, 25 | sylbi 114 | . . . . . 6 |
27 | opelxp1 4377 | . . . . . 6 | |
28 | 26, 27 | syl 14 | . . . . 5 |
29 | 28 | exlimiv 1489 | . . . 4 |
30 | 18, 19, 29 | 3syl 17 | . . 3 |
31 | 30 | con3i 562 | . 2 |
32 | 31 | eq0rdv 3261 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wceq 1243 wex 1381 wcel 1393 wsb 1645 weu 1900 cab 2026 cvv 2557 cin 2916 c0 3224 csn 3375 cop 3378 cuni 3580 class class class wbr 3764 cxp 4343 cres 4347 cio 4865 cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-res 4357 df-iota 4867 df-fv 4910 |
This theorem is referenced by: (None) |
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