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| 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 4400 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
| 2 | df-pss 3971 | . . 3 ⊢ (∅ ⊊ 𝐴 ↔ (∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴)) | |
| 3 | 1, 2 | mpbiran 709 | . 2 ⊢ (∅ ⊊ 𝐴 ↔ ∅ ≠ 𝐴) | 
| 4 | necom 2994 | . 2 ⊢ (∅ ≠ 𝐴 ↔ 𝐴 ≠ ∅) | |
| 5 | 3, 4 | bitri 275 | 1 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ≠ wne 2940 ⊆ wss 3951 ⊊ wpss 3952 ∅c0 4333 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-dif 3954 df-ss 3968 df-pss 3971 df-nul 4334 | 
| This theorem is referenced by: php 9247 phpOLD 9259 zornn0g 10545 prn0 11029 genpn0 11043 nqpr 11054 ltexprlem5 11080 reclem2pr 11088 suplem1pr 11092 alexsubALTlem4 24058 bj-2upln0 37024 bj-2upln1upl 37025 0pssin 43784 | 
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