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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 4429 | . 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} ↔ ∀𝑥((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝐴 ⊆ 𝑥)) | |
2 | simpl 472 | . 2 ⊢ ((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝐴 ⊆ 𝑥) | |
3 | 1, 2 | mpgbir 1717 | 1 ⊢ 𝐴 ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 {cab 2596 ⊆ wss 3540 ∩ cint 4410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-in 3547 df-ss 3554 df-int 4411 |
This theorem is referenced by: tcid 8498 trclfvlb 13597 trclun 13603 dmtrcl 36953 rntrcl 36954 dfrtrcl5 36955 |
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