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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssdf2 | Structured version Visualization version GIF version |
Description: A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
ssdf2.p | ⊢ Ⅎ𝑥𝜑 |
ssdf2.a | ⊢ Ⅎ𝑥𝐴 |
ssdf2.b | ⊢ Ⅎ𝑥𝐵 |
ssdf2.x | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) |
Ref | Expression |
---|---|
ssdf2 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdf2.p | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | ssdf2.a | . 2 ⊢ Ⅎ𝑥𝐴 | |
3 | ssdf2.b | . 2 ⊢ Ⅎ𝑥𝐵 | |
4 | ssdf2.x | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) | |
5 | 4 | ex 412 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
6 | 1, 2, 3, 5 | ssrd 4000 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 ⊆ wss 3963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-11 2155 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-nf 1781 df-clel 2814 df-nfc 2890 df-ss 3980 |
This theorem is referenced by: supminfxr2 45419 fsupdm 46798 finfdm 46802 |
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