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Mirrors > Home > MPE Home > Th. List > 0pss | Structured version Visualization version GIF version |
Description: The null set is a proper subset of any nonempty set. (Contributed by NM, 27-Feb-1996.) |
Ref | Expression |
---|---|
0pss | ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4103 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
2 | df-pss 3719 | . . 3 ⊢ (∅ ⊊ 𝐴 ↔ (∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴)) | |
3 | 1, 2 | mpbiran 991 | . 2 ⊢ (∅ ⊊ 𝐴 ↔ ∅ ≠ 𝐴) |
4 | necom 2973 | . 2 ⊢ (∅ ≠ 𝐴 ↔ 𝐴 ≠ ∅) | |
5 | 3, 4 | bitri 264 | 1 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ≠ wne 2920 ⊆ wss 3703 ⊊ wpss 3704 ∅c0 4046 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-v 3330 df-dif 3706 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 |
This theorem is referenced by: php 8297 zornn0g 9490 prn0 9974 genpn0 9988 nqpr 9999 ltexprlem5 10025 reclem2pr 10033 suplem1pr 10037 alexsubALTlem4 22026 bj-2upln0 33288 bj-2upln1upl 33289 0pssin 38535 |
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