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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 4106. (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 4106 | . . 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 696 | 1 ⊢ (𝜑 → 𝐴 ⊊ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2098 ⊆ wss 3943 ⊊ wpss 3944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-cleq 2718 df-clel 2804 df-ne 2935 df-pss 3962 |
This theorem is referenced by: canth4 10644 mrieqv2d 17592 symgpssefmnd 19315 symggen 19390 pgpfac1lem1 19996 pgpfaclem2 20004 |
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