![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ssnpss | Structured version Visualization version GIF version |
Description: Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
ssnpss | ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ⊊ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfpss3 4087 | . . 3 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ 𝐵)) | |
2 | 1 | simprbi 498 | . 2 ⊢ (𝐵 ⊊ 𝐴 → ¬ 𝐴 ⊆ 𝐵) |
3 | 2 | con2i 139 | 1 ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ⊊ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ⊆ wss 3949 ⊊ wpss 3950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-v 3477 df-in 3956 df-ss 3966 df-pss 3968 |
This theorem is referenced by: npss0 4446 sorpssuni 7722 sorpssint 7723 suplem2pr 11048 symgvalstruct 19264 symgvalstructOLD 19265 lsppratlem6 20765 atcvati 31639 finxpreclem3 36274 lsatcvat 37920 |
Copyright terms: Public domain | W3C validator |