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Mirrors > Home > ILE Home > Th. List > inopn | Unicode version |
Description: The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.) |
Ref | Expression |
---|---|
inopn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istopg 13951 |
. . . . 5
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2 | 1 | ibi 176 |
. . . 4
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3 | 2 | simprd 114 |
. . 3
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4 | ineq1 3344 |
. . . . 5
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5 | 4 | eleq1d 2258 |
. . . 4
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6 | ineq2 3345 |
. . . . 5
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7 | 6 | eleq1d 2258 |
. . . 4
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8 | 5, 7 | rspc2v 2869 |
. . 3
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9 | 3, 8 | syl5com 29 |
. 2
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10 | 9 | 3impib 1203 |
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-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-sep 4136 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2754 df-in 3150 df-ss 3157 df-pw 3592 df-top 13950 |
This theorem is referenced by: tgclb 14017 topbas 14019 difopn 14060 uncld 14065 ntrin 14076 innei 14115 restopnb 14133 cnptoprest 14191 txcnp 14223 txcnmpt 14225 mopnin 14439 |
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