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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 415 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
6 | 1, 2, 3, 5 | ssrd 3972 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2961 ⊆ wss 3936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-in 3943 df-ss 3952 |
This theorem is referenced by: supminfxr2 41765 |
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