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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 5291 | . . . . . . . 8 | |
2 | elmapg 6523 | . . . . . . . 8 | |
3 | 1, 2 | syl5ibr 155 | . . . . . . 7 |
4 | ne0i 3339 | . . . . . . 7 | |
5 | 3, 4 | syl6 33 | . . . . . 6 |
6 | 5 | exlimdv 1775 | . . . . 5 |
7 | 6 | necon2bd 2343 | . . . 4 |
8 | notm0 3353 | . . . 4 | |
9 | 7, 8 | syl6ib 160 | . . 3 |
10 | f0 5283 | . . . . . . 7 | |
11 | feq2 5226 | . . . . . . 7 | |
12 | 10, 11 | mpbiri 167 | . . . . . 6 |
13 | elmapg 6523 | . . . . . 6 | |
14 | 12, 13 | syl5ibr 155 | . . . . 5 |
15 | ne0i 3339 | . . . . 5 | |
16 | 14, 15 | syl6 33 | . . . 4 |
17 | 16 | necon2d 2344 | . . 3 |
18 | 9, 17 | jcad 305 | . 2 |
19 | oveq1 5749 | . . 3 | |
20 | map0b 6549 | . . 3 | |
21 | 19, 20 | sylan9eq 2170 | . 2 |
22 | 18, 21 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1316 wex 1453 wcel 1465 wne 2285 c0 3333 csn 3497 cxp 4507 wf 5089 (class class class)co 5742 cmap 6510 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-map 6512 |
This theorem is referenced by: map0 6551 |
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