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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 4086 | . . 3 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ 𝐵)) | |
2 | 1 | simprbi 497 | . 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 3948 ⊊ wpss 3949 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-v 3476 df-in 3955 df-ss 3965 df-pss 3967 |
This theorem is referenced by: npss0 4445 sorpssuni 7721 sorpssint 7722 suplem2pr 11047 symgvalstruct 19263 symgvalstructOLD 19264 lsppratlem6 20764 atcvati 31634 finxpreclem3 36269 lsatcvat 37915 |
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