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Mirrors > Home > MPE Home > Th. List > ssmin | Structured version Visualization version GIF version |
Description: Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.) |
Ref | Expression |
---|---|
ssmin | ⊢ 𝐴 ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssintab 4926 | . 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} ↔ ∀𝑥((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝐴 ⊆ 𝑥)) | |
2 | simpl 483 | . 2 ⊢ ((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝐴 ⊆ 𝑥) | |
3 | 1, 2 | mpgbir 1801 | 1 ⊢ 𝐴 ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 {cab 2713 ⊆ wss 3910 ∩ cint 4907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-v 3447 df-in 3917 df-ss 3927 df-int 4908 |
This theorem is referenced by: tcid 9674 trclfvlb 14892 trclun 14898 dmtrcl 41880 rntrcl 41881 dfrtrcl5 41882 |
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