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Mirrors > Home > MPE Home > Th. List > ssnelpssd | Structured version Visualization version GIF version |
Description: Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 4002. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
ssnelpssd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
ssnelpssd.2 | ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
ssnelpssd.3 | ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) |
Ref | Expression |
---|---|
ssnelpssd | ⊢ (𝜑 → 𝐴 ⊊ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssnelpssd.2 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐵) | |
2 | ssnelpssd.3 | . 2 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) | |
3 | ssnelpssd.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
4 | ssnelpss 4002 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐴 ⊊ 𝐵)) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐴 ⊊ 𝐵)) |
6 | 1, 2, 5 | mp2and 699 | 1 ⊢ (𝜑 → 𝐴 ⊊ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∈ wcel 2114 ⊆ wss 3843 ⊊ wpss 3844 |
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 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1787 df-cleq 2730 df-clel 2811 df-ne 2935 df-pss 3862 |
This theorem is referenced by: canth4 10147 mrieqv2d 17013 symgpssefmnd 18642 symggen 18716 pgpfac1lem1 19315 pgpfaclem2 19323 |
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