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Mirrors > Home > MPE Home > Th. List > tgdom | Structured version Visualization version Unicode version |
Description: A space has no more open sets than subsets of a basis. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 9-Apr-2015.) |
Ref | Expression |
---|---|
tgdom |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwexg 4587 |
. 2
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2 | inss1 3652 |
. . . . 5
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3 | vex 3048 |
. . . . . . . 8
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4 | 3 | pwex 4586 |
. . . . . . 7
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5 | 4 | inex2 4545 |
. . . . . 6
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6 | 5 | elpw 3957 |
. . . . 5
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7 | 2, 6 | mpbir 213 |
. . . 4
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8 | 7 | a1i 11 |
. . 3
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9 | unieq 4206 |
. . . . . . 7
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10 | 9 | adantl 468 |
. . . . . 6
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11 | eltg4i 19975 |
. . . . . . 7
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12 | 11 | ad2antrr 732 |
. . . . . 6
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13 | eltg4i 19975 |
. . . . . . 7
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14 | 13 | ad2antlr 733 |
. . . . . 6
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15 | 10, 12, 14 | 3eqtr4d 2495 |
. . . . 5
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16 | 15 | ex 436 |
. . . 4
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17 | pweq 3954 |
. . . . 5
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18 | 17 | ineq2d 3634 |
. . . 4
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19 | 16, 18 | impbid1 207 |
. . 3
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20 | 8, 19 | dom2 7612 |
. 2
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21 | 1, 20 | syl 17 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-reu 2744 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-op 3975 df-uni 4199 df-iun 4280 df-br 4403 df-opab 4462 df-mpt 4463 df-id 4749 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-dom 7571 df-topgen 15342 |
This theorem is referenced by: 2ndcredom 20465 kelac2lem 35922 |
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