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Mirrors > Home > NFE Home > Th. List > map0 | 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 set.mm contributors, 10-Dec-2003.) (Revised by set.mm contributors, 17-May-2007.) |
Ref | Expression |
---|---|
map0.1 | |
map0.2 |
Ref | Expression |
---|---|
map0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | map0.1 | . . . . . 6 | |
2 | map0.2 | . . . . . 6 | |
3 | 1, 2 | mapval 6011 | . . . . 5 |
4 | 3 | eqeq1i 2360 | . . . 4 |
5 | snssi 3852 | . . . . . . . 8 | |
6 | vex 2862 | . . . . . . . . . 10 | |
7 | 6 | fconst 5250 | . . . . . . . . 9 |
8 | fss 5230 | . . . . . . . . 9 | |
9 | 7, 8 | mpan 651 | . . . . . . . 8 |
10 | snex 4111 | . . . . . . . . . 10 | |
11 | 2, 10 | xpex 5115 | . . . . . . . . 9 |
12 | feq1 5210 | . . . . . . . . 9 | |
13 | 11, 12 | spcev 2946 | . . . . . . . 8 |
14 | 5, 9, 13 | 3syl 18 | . . . . . . 7 |
15 | 14 | exlimiv 1634 | . . . . . 6 |
16 | n0 3559 | . . . . . 6 | |
17 | abn0 3568 | . . . . . 6 | |
18 | 15, 16, 17 | 3imtr4i 257 | . . . . 5 |
19 | 18 | necon4i 2576 | . . . 4 |
20 | 4, 19 | sylbi 187 | . . 3 |
21 | 1 | map0e 6023 | . . . . . 6 |
22 | 0ex 4110 | . . . . . . . 8 | |
23 | 22 | snid 3760 | . . . . . . 7 |
24 | ne0i 3556 | . . . . . . 7 | |
25 | 23, 24 | ax-mp 5 | . . . . . 6 |
26 | 21, 25 | eqnetri 2533 | . . . . 5 |
27 | oveq2 5531 | . . . . . 6 | |
28 | 27 | neeq1d 2529 | . . . . 5 |
29 | 26, 28 | mpbiri 224 | . . . 4 |
30 | 29 | necon2i 2563 | . . 3 |
31 | 20, 30 | jca 518 | . 2 |
32 | oveq1 5530 | . . 3 | |
33 | 2 | map0b 6024 | . . 3 |
34 | 32, 33 | sylan9eq 2405 | . 2 |
35 | 31, 34 | impbii 180 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 176 wa 358 wex 1541 wceq 1642 wcel 1710 cab 2339 wne 2516 cvv 2859 wss 3257 c0 3550 csn 3737 cxp 4770 wf 4777 (class class class)co 5525 cmap 5999 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-si 4728 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-fv 4795 df-2nd 4797 df-ov 5526 df-oprab 5528 df-mpt2 5654 df-txp 5736 df-ins2 5750 df-ins3 5752 df-image 5754 df-ins4 5756 df-si3 5758 df-funs 5760 df-map 6001 |
This theorem is referenced by: (None) |
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