![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4304 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
2 | df-pss 3900 | . . 3 ⊢ (∅ ⊊ 𝐴 ↔ (∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴)) | |
3 | 1, 2 | mpbiran 708 | . 2 ⊢ (∅ ⊊ 𝐴 ↔ ∅ ≠ 𝐴) |
4 | necom 3040 | . 2 ⊢ (∅ ≠ 𝐴 ↔ 𝐴 ≠ ∅) | |
5 | 3, 4 | bitri 278 | 1 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ≠ wne 2987 ⊆ wss 3881 ⊊ wpss 3882 ∅c0 4243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 |
This theorem is referenced by: php 8685 zornn0g 9916 prn0 10400 genpn0 10414 nqpr 10425 ltexprlem5 10451 reclem2pr 10459 suplem1pr 10463 alexsubALTlem4 22655 bj-2upln0 34459 bj-2upln1upl 34460 0pssin 40472 |
Copyright terms: Public domain | W3C validator |