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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 3985 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 Ⅎwnf 1778 ∈ wcel 2099 Ⅎwnfc 2879 ⊆ wss 3947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-11 2147 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-v 3473 df-in 3954 df-ss 3964 |
This theorem is referenced by: supminfxr2 44851 fsupdm 46230 finfdm 46234 |
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