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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 5131 | . . . . . . . . . 10 | |
2 | df-iota 5088 | . . . . . . . . . 10 | |
3 | 1, 2 | eqtri 2160 | . . . . . . . . 9 |
4 | 3 | eleq2i 2206 | . . . . . . . 8 |
5 | eluni 3739 | . . . . . . . 8 | |
6 | 4, 5 | bitri 183 | . . . . . . 7 |
7 | exsimpr 1597 | . . . . . . 7 | |
8 | 6, 7 | sylbi 120 | . . . . . 6 |
9 | df-clab 2126 | . . . . . . . 8 | |
10 | nfv 1508 | . . . . . . . . 9 | |
11 | sneq 3538 | . . . . . . . . . 10 | |
12 | 11 | eqeq2d 2151 | . . . . . . . . 9 |
13 | 10, 12 | sbie 1764 | . . . . . . . 8 |
14 | 9, 13 | bitri 183 | . . . . . . 7 |
15 | 14 | exbii 1584 | . . . . . 6 |
16 | 8, 15 | sylib 121 | . . . . 5 |
17 | euabsn2 3592 | . . . . 5 | |
18 | 16, 17 | sylibr 133 | . . . 4 |
19 | euex 2029 | . . . 4 | |
20 | df-br 3930 | . . . . . . . 8 | |
21 | df-res 4551 | . . . . . . . . 9 | |
22 | 21 | eleq2i 2206 | . . . . . . . 8 |
23 | 20, 22 | bitri 183 | . . . . . . 7 |
24 | elin 3259 | . . . . . . . 8 | |
25 | 24 | simprbi 273 | . . . . . . 7 |
26 | 23, 25 | sylbi 120 | . . . . . 6 |
27 | opelxp1 4573 | . . . . . 6 | |
28 | 26, 27 | syl 14 | . . . . 5 |
29 | 28 | exlimiv 1577 | . . . 4 |
30 | 18, 19, 29 | 3syl 17 | . . 3 |
31 | 30 | con3i 621 | . 2 |
32 | 31 | eq0rdv 3407 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1331 wex 1468 wcel 1480 wsb 1735 weu 1999 cab 2125 cvv 2686 cin 3070 c0 3363 csn 3527 cop 3530 cuni 3736 class class class wbr 3929 cxp 4537 cres 4541 cio 5086 cfv 5123 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-res 4551 df-iota 5088 df-fv 5131 |
This theorem is referenced by: (None) |
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