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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 3924 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
2 | df-pss 3556 | . . 3 ⊢ (∅ ⊊ 𝐴 ↔ (∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴)) | |
3 | 1, 2 | mpbiran 955 | . 2 ⊢ (∅ ⊊ 𝐴 ↔ ∅ ≠ 𝐴) |
4 | necom 2835 | . 2 ⊢ (∅ ≠ 𝐴 ↔ 𝐴 ≠ ∅) | |
5 | 3, 4 | bitri 263 | 1 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ≠ wne 2780 ⊆ wss 3540 ⊊ wpss 3541 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 |
This theorem is referenced by: php 8029 zornn0g 9210 prn0 9690 genpn0 9704 nqpr 9715 ltexprlem5 9741 reclem2pr 9749 suplem1pr 9753 alexsubALTlem4 21664 bj-2upln0 32204 bj-2upln1upl 32205 0pssin 37084 |
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