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Mirrors > Home > ILE Home > Th. List > map0g | Unicode version |
Description: Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
map0g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconst6g 5321 | . . . . . . . 8 | |
2 | elmapg 6555 | . . . . . . . 8 | |
3 | 1, 2 | syl5ibr 155 | . . . . . . 7 |
4 | ne0i 3369 | . . . . . . 7 | |
5 | 3, 4 | syl6 33 | . . . . . 6 |
6 | 5 | exlimdv 1791 | . . . . 5 |
7 | 6 | necon2bd 2366 | . . . 4 |
8 | notm0 3383 | . . . 4 | |
9 | 7, 8 | syl6ib 160 | . . 3 |
10 | f0 5313 | . . . . . . 7 | |
11 | feq2 5256 | . . . . . . 7 | |
12 | 10, 11 | mpbiri 167 | . . . . . 6 |
13 | elmapg 6555 | . . . . . 6 | |
14 | 12, 13 | syl5ibr 155 | . . . . 5 |
15 | ne0i 3369 | . . . . 5 | |
16 | 14, 15 | syl6 33 | . . . 4 |
17 | 16 | necon2d 2367 | . . 3 |
18 | 9, 17 | jcad 305 | . 2 |
19 | oveq1 5781 | . . 3 | |
20 | map0b 6581 | . . 3 | |
21 | 19, 20 | sylan9eq 2192 | . 2 |
22 | 18, 21 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 wne 2308 c0 3363 csn 3527 cxp 4537 wf 5119 (class class class)co 5774 cmap 6542 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 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-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-map 6544 |
This theorem is referenced by: map0 6583 |
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