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Mirrors > Home > MPE Home > Th. List > 0pss | Structured version Visualization version Unicode version |
Description: The null set is a proper subset of any nonempty set. (Contributed by NM, 27-Feb-1996.) |
Ref | Expression |
---|---|
0pss |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 3774 |
. . 3
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2 | df-pss 3431 |
. . 3
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3 | 1, 2 | mpbiran 934 |
. 2
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4 | necom 2688 |
. 2
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5 | 3, 4 | bitri 257 |
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 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 |
This theorem depends on definitions: df-bi 190 df-an 377 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-v 3058 df-dif 3418 df-in 3422 df-ss 3429 df-pss 3431 df-nul 3743 |
This theorem is referenced by: php 7781 zornn0g 8960 prn0 9439 genpn0 9453 nqpr 9464 ltexprlem5 9490 reclem2pr 9498 suplem1pr 9502 alexsubALTlem4 21113 bj-2upln0 31661 bj-2upln1upl 31662 0pssin 36409 |
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