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Mirrors > Home > NFE Home > Th. List > f1o00 | Unicode version |
Description: One-to-one onto mapping of the empty set. (Contributed by set.mm contributors, 15-Apr-1998.) |
Ref | Expression |
---|---|
f1o00 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff1o4 5295 |
. 2
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2 | fn0 5203 |
. . . . . 6
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3 | 2 | biimpi 186 |
. . . . 5
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4 | 3 | adantr 451 |
. . . 4
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5 | dm0 4919 |
. . . . 5
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6 | cnveq 4887 |
. . . . . . . . . 10
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7 | cnv0 5032 |
. . . . . . . . . 10
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8 | 6, 7 | syl6eq 2401 |
. . . . . . . . 9
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9 | 2, 8 | sylbi 187 |
. . . . . . . 8
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10 | 9 | fneq1d 5176 |
. . . . . . 7
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11 | 10 | biimpa 470 |
. . . . . 6
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12 | fndm 5183 |
. . . . . 6
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13 | 11, 12 | syl 15 |
. . . . 5
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14 | 5, 13 | syl5reqr 2400 |
. . . 4
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15 | 4, 14 | jca 518 |
. . 3
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16 | 2 | biimpri 197 |
. . . . 5
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17 | 16 | adantr 451 |
. . . 4
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18 | eqid 2353 |
. . . . . 6
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19 | fn0 5203 |
. . . . . 6
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20 | 18, 19 | mpbir 200 |
. . . . 5
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21 | 8 | fneq1d 5176 |
. . . . . 6
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22 | fneq2 5175 |
. . . . . 6
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23 | 21, 22 | sylan9bb 680 |
. . . . 5
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24 | 20, 23 | mpbiri 224 |
. . . 4
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25 | 17, 24 | jca 518 |
. . 3
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26 | 15, 25 | impbii 180 |
. 2
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27 | 1, 26 | bitri 240 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 4079 ax-xp 4080 ax-cnv 4081 ax-1c 4082 ax-sset 4083 ax-si 4084 ax-ins2 4085 ax-ins3 4086 ax-typlower 4087 ax-sn 4088 |
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 2479 df-ne 2519 df-ral 2620 df-rex 2621 df-reu 2622 df-rmo 2623 df-rab 2624 df-v 2862 df-sbc 3048 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-symdif 3217 df-ss 3260 df-pss 3262 df-nul 3552 df-if 3664 df-pw 3725 df-sn 3742 df-pr 3743 df-uni 3893 df-int 3928 df-opk 4059 df-1c 4137 df-pw1 4138 df-uni1 4139 df-xpk 4186 df-cnvk 4187 df-ins2k 4188 df-ins3k 4189 df-imak 4190 df-cok 4191 df-p6 4192 df-sik 4193 df-ssetk 4194 df-imagek 4195 df-idk 4196 df-iota 4340 df-0c 4378 df-addc 4379 df-nnc 4380 df-fin 4381 df-lefin 4441 df-ltfin 4442 df-ncfin 4443 df-tfin 4444 df-evenfin 4445 df-oddfin 4446 df-sfin 4447 df-spfin 4448 df-phi 4566 df-op 4567 df-proj1 4568 df-proj2 4569 df-opab 4624 df-br 4641 df-co 4727 df-ima 4728 df-cnv 4786 df-rn 4787 df-dm 4788 df-fun 4790 df-fn 4791 df-f 4792 df-f1 4793 df-fo 4794 df-f1o 4795 |
This theorem is referenced by: fo00 5319 en0 6043 |
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