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Mirrors > Home > MPE Home > Th. List > ssrd | Structured version Visualization version GIF version |
Description: Deduction based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.) |
Ref | Expression |
---|---|
ssrd.0 | ⊢ Ⅎ𝑥𝜑 |
ssrd.1 | ⊢ Ⅎ𝑥𝐴 |
ssrd.2 | ⊢ Ⅎ𝑥𝐵 |
ssrd.3 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
Ref | Expression |
---|---|
ssrd | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrd.0 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | ssrd.3 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | alrimi 2210 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
4 | ssrd.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
5 | ssrd.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
6 | 4, 5 | dfss2f 3916 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
7 | 3, 6 | sylibr 233 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 Ⅎwnf 1790 ∈ wcel 2110 Ⅎwnfc 2889 ⊆ wss 3892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-11 2158 ax-12 2175 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-v 3433 df-in 3899 df-ss 3909 |
This theorem is referenced by: neiptopnei 22281 rabss3d 30857 topdifinffinlem 35514 relowlssretop 35530 ralssiun 35574 sticksstones1 40099 sticksstones11 40109 ssdf2 42660 ssfiunibd 42819 stoweidlem52 43564 stoweidlem59 43571 |
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