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Mirrors > Home > ILE Home > Th. List > 0opn | Unicode version |
Description: The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.) |
Ref | Expression |
---|---|
0opn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uni0 3821 | . 2 | |
2 | 0ss 3452 | . . 3 | |
3 | uniopn 12758 | . . 3 | |
4 | 2, 3 | mpan2 423 | . 2 |
5 | 1, 4 | eqeltrrid 2258 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 2141 wss 3121 c0 3414 cuni 3794 ctop 12754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-sep 4105 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-uni 3795 df-top 12755 |
This theorem is referenced by: 0ntop 12764 topgele 12786 istps 12789 topontopn 12794 tgclb 12824 en1top 12836 topcld 12868 ntr0 12893 0nei 12925 restrcl 12926 rest0 12938 mopn0 13247 |
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