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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 2207 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
4 | ssrd.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
5 | ssrd.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
6 | 4, 5 | dfss2f 3939 | . 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 1786 ∈ wcel 2107 Ⅎwnfc 2888 ⊆ wss 3915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-v 3450 df-in 3922 df-ss 3932 |
This theorem is referenced by: rabss3d 4044 neiptopnei 22499 topdifinffinlem 35847 relowlssretop 35863 ralssiun 35907 sticksstones1 40583 sticksstones11 40593 ssdf2 43425 ssfiunibd 43617 stoweidlem52 44367 stoweidlem59 44374 |
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