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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 2211 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
4 | ssrd.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
5 | ssrd.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
6 | 4, 5 | dfssf 3986 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
7 | 3, 6 | sylibr 234 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 ⊆ wss 3963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-11 2155 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-nf 1781 df-clel 2814 df-nfc 2890 df-ss 3980 |
This theorem is referenced by: rabss3d 4091 neiptopnei 23156 topdifinffinlem 37330 relowlssretop 37346 ralssiun 37390 sticksstones1 42128 sticksstones11 42138 ssdf2 45081 ssfiunibd 45260 stoweidlem52 46008 stoweidlem59 46015 |
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