![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4072. (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 4072 | . . 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 698 | 1 ⊢ (𝜑 → 𝐴 ⊊ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∈ wcel 2107 ⊆ wss 3911 ⊊ wpss 3912 |
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-ex 1783 df-cleq 2725 df-clel 2811 df-ne 2941 df-pss 3930 |
This theorem is referenced by: canth4 10588 mrieqv2d 17524 symgpssefmnd 19182 symggen 19257 pgpfac1lem1 19858 pgpfaclem2 19866 |
Copyright terms: Public domain | W3C validator |