![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ssdf | Structured version Visualization version GIF version |
Description: A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
ssdf.1 | ⊢ Ⅎ𝑥𝜑 |
ssdf.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) |
Ref | Expression |
---|---|
ssdf | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdf.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | ssdf.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) | |
3 | 2 | ex 449 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
4 | 1, 3 | ralrimi 3087 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
5 | dfss3 3725 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) | |
6 | 4, 5 | sylibr 224 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 Ⅎwnf 1849 ∈ wcel 2131 ∀wral 3042 ⊆ wss 3707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-clab 2739 df-cleq 2745 df-clel 2748 df-ral 3047 df-in 3714 df-ss 3721 |
This theorem is referenced by: ssd 39743 smfaddlem2 41470 smfadd 41471 smfmullem4 41499 smfmul 41500 smflimsuplem4 41527 |
Copyright terms: Public domain | W3C validator |