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Mirrors > Home > ILE Home > Th. List > 0elnn | Unicode version |
Description: A natural number is either the empty set or has the empty set as an element. (Contributed by Jim Kingdon, 23-Aug-2019.) |
Ref | Expression |
---|---|
0elnn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2184 |
. . 3
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2 | eleq2 2241 |
. . 3
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3 | 1, 2 | orbi12d 793 |
. 2
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4 | eqeq1 2184 |
. . 3
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5 | eleq2 2241 |
. . 3
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6 | 4, 5 | orbi12d 793 |
. 2
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7 | eqeq1 2184 |
. . 3
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8 | eleq2 2241 |
. . 3
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9 | 7, 8 | orbi12d 793 |
. 2
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10 | eqeq1 2184 |
. . 3
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11 | eleq2 2241 |
. . 3
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12 | 10, 11 | orbi12d 793 |
. 2
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13 | eqid 2177 |
. . 3
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14 | 13 | orci 731 |
. 2
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15 | 0ex 4130 |
. . . . . . 7
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16 | 15 | sucid 4417 |
. . . . . 6
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17 | suceq 4402 |
. . . . . 6
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18 | 16, 17 | eleqtrrid 2267 |
. . . . 5
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19 | 18 | a1i 9 |
. . . 4
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20 | sssucid 4415 |
. . . . . 6
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21 | 20 | a1i 9 |
. . . . 5
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22 | 21 | sseld 3154 |
. . . 4
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23 | 19, 22 | jaod 717 |
. . 3
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24 | olc 711 |
. . 3
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25 | 23, 24 | syl6 33 |
. 2
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26 | 3, 6, 9, 12, 14, 25 | finds 4599 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-iinf 4587 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-uni 3810 df-int 3845 df-suc 4371 df-iom 4590 |
This theorem is referenced by: nn0eln0 4619 nnsucsssuc 6492 nntri3or 6493 nnm00 6530 ssfilem 6874 diffitest 6886 fiintim 6927 enumct 7113 nnnninfeq 7125 elni2 7312 enq0tr 7432 bj-charfunr 14482 |
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