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Mirrors > Home > MPE Home > Th. List > fi0 | Structured version Visualization version Unicode version |
Description: The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.) |
Ref | Expression |
---|---|
fi0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4528 |
. . 3
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2 | fival 7944 |
. . 3
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3 | 1, 2 | ax-mp 5 |
. 2
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4 | vprc 4534 |
. . . 4
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5 | id 22 |
. . . . . . 7
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6 | inss1 3643 |
. . . . . . . . . . 11
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7 | 6 | sseli 3414 |
. . . . . . . . . 10
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8 | elpwi 3951 |
. . . . . . . . . 10
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9 | ss0 3768 |
. . . . . . . . . 10
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10 | 7, 8, 9 | 3syl 18 |
. . . . . . . . 9
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11 | 10 | inteqd 4231 |
. . . . . . . 8
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12 | int0 4240 |
. . . . . . . 8
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13 | 11, 12 | syl6eq 2521 |
. . . . . . 7
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14 | 5, 13 | sylan9eqr 2527 |
. . . . . 6
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15 | 14 | rexlimiva 2868 |
. . . . 5
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16 | vex 3034 |
. . . . 5
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17 | 15, 16 | syl6eqelr 2558 |
. . . 4
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18 | 4, 17 | mto 181 |
. . 3
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19 | 18 | abf 3772 |
. 2
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20 | 3, 19 | eqtri 2493 |
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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-rab 2765 df-v 3033 df-sbc 3256 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-op 3966 df-uni 4191 df-int 4227 df-br 4396 df-opab 4455 df-mpt 4456 df-id 4754 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-iota 5553 df-fun 5591 df-fv 5597 df-fi 7943 |
This theorem is referenced by: fieq0 7953 firest 15409 restbas 20251 |
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