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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssd | Structured version Visualization version GIF version |
Description: A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
ssd.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) |
Ref | Expression |
---|---|
ssd | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1830 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | ssd.1 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) | |
3 | 1, 2 | ssdf 38273 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ⊆ wss 3540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-ral 2901 df-in 3547 df-ss 3554 |
This theorem is referenced by: icomnfinre 38626 fnlimfvre 38741 allbutfifvre 38742 rrxsnicc 39196 meaiuninclem 39373 meaiininclem 39376 borelmbl 39526 smflimlem1 39657 smflimlem2 39658 smfpimbor1lem1 39683 smfpimbor1lem2 39684 |
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