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Mirrors > Home > ILE Home > Th. List > 0ct | Unicode version |
Description: The empty set is countable. Remark of [BauerSwan], p. 14:3 which also has the definition of countable used here. (Contributed by Jim Kingdon, 13-Mar-2023.) |
Ref | Expression |
---|---|
0ct |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0lt1o 6455 |
. . . . 5
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2 | djurcl 7065 |
. . . . 5
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3 | 1, 2 | ax-mp 5 |
. . . 4
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4 | 3 | fconst6 5427 |
. . 3
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5 | peano1 4605 |
. . . . 5
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6 | rex0 3452 |
. . . . . . . . 9
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7 | djur 7082 |
. . . . . . . . . . 11
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8 | 7 | biimpi 120 |
. . . . . . . . . 10
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9 | 8 | ord 725 |
. . . . . . . . 9
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10 | 6, 9 | mpi 15 |
. . . . . . . 8
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11 | df1o2 6444 |
. . . . . . . . 9
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12 | 11 | rexeqi 2688 |
. . . . . . . 8
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13 | 10, 12 | sylib 122 |
. . . . . . 7
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14 | 0ex 4142 |
. . . . . . . 8
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15 | fveq2 5527 |
. . . . . . . . 9
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16 | 15 | eqeq2d 2199 |
. . . . . . . 8
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17 | 14, 16 | rexsn 3648 |
. . . . . . 7
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18 | 13, 17 | sylib 122 |
. . . . . 6
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19 | 3 | elexi 2761 |
. . . . . . . 8
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20 | 19 | fvconst2 5745 |
. . . . . . 7
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21 | 5, 20 | ax-mp 5 |
. . . . . 6
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22 | 18, 21 | eqtr4di 2238 |
. . . . 5
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23 | fveq2 5527 |
. . . . . 6
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24 | 23 | rspceeqv 2871 |
. . . . 5
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25 | 5, 22, 24 | sylancr 414 |
. . . 4
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26 | 25 | rgen 2540 |
. . 3
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27 | dffo3 5676 |
. . 3
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28 | 4, 26, 27 | mpbir2an 943 |
. 2
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29 | omex 4604 |
. . . 4
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30 | 19 | snex 4197 |
. . . 4
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31 | 29, 30 | xpex 4753 |
. . 3
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32 | foeq1 5446 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
33 | 31, 32 | spcev 2844 |
. 2
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34 | 28, 33 | ax-mp 5 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-iinf 4599 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-iord 4378 df-on 4380 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-1st 6155 df-2nd 6156 df-1o 6431 df-dju 7051 df-inl 7060 df-inr 7061 |
This theorem is referenced by: enumct 7128 |
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