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| 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 3988 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 ⊆ wss 3951 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-11 2157 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-clel 2816 df-nfc 2892 df-ss 3968 | 
| This theorem is referenced by: supminfxr2 45480 fsupdm 46857 finfdm 46861 | 
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