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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 2143 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
4 | ssrd.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
5 | ssrd.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
6 | 4, 5 | dfss2f 3849 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
7 | 3, 6 | sylibr 226 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1505 Ⅎwnf 1746 ∈ wcel 2050 Ⅎwnfc 2916 ⊆ wss 3829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-in 3836 df-ss 3843 |
This theorem is referenced by: neiptopnei 21444 rabss3d 30052 topdifinffinlem 34076 relowlssretop 34092 ralssiun 34135 ssdf2 40838 ssfiunibd 41011 stoweidlem52 41774 stoweidlem59 41781 |
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