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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 3975 | . . 3 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ 𝐵)) | |
2 | 1 | simprbi 500 | . 2 ⊢ (𝐵 ⊊ 𝐴 → ¬ 𝐴 ⊆ 𝐵) |
3 | 2 | con2i 141 | 1 ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ⊊ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ⊆ wss 3841 ⊊ wpss 3842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-ne 2935 df-v 3399 df-in 3848 df-ss 3858 df-pss 3860 |
This theorem is referenced by: npss0 4332 sorpssuni 7470 sorpssint 7471 suplem2pr 10546 symgvalstruct 18636 lsppratlem6 20036 atcvati 30313 finxpreclem3 35176 lsatcvat 36676 |
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