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Mirrors > Home > MPE Home > Th. List > ssrd | Structured version Visualization version GIF version |
Description: Deduction rule 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 2082 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
4 | ssrd.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
5 | ssrd.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
6 | 4, 5 | dfss2f 3594 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
7 | 3, 6 | sylibr 224 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1481 Ⅎwnf 1708 ∈ wcel 1990 Ⅎwnfc 2751 ⊆ wss 3574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-in 3581 df-ss 3588 |
This theorem is referenced by: eqrdOLD 3623 neiptopnei 20936 rabss3d 29351 topdifinffinlem 33195 relowlssretop 33211 ssdf2 39331 ssfiunibd 39523 stoweidlem52 40269 stoweidlem59 40276 |
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