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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 4970 | . 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} ↔ ∀𝑥((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝐴 ⊆ 𝑥)) | |
2 | simpl 482 | . 2 ⊢ ((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝐴 ⊆ 𝑥) | |
3 | 1, 2 | mpgbir 1796 | 1 ⊢ 𝐴 ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 {cab 2712 ⊆ wss 3963 ∩ cint 4951 |
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-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-v 3480 df-ss 3980 df-int 4952 |
This theorem is referenced by: tcid 9777 trclfvlb 15044 trclun 15050 dmtrcl 43617 rntrcl 43618 dfrtrcl5 43619 |
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