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Mirrors > Home > MPE Home > Th. List > intmin3 | Structured version Visualization version GIF version |
Description: Under subset ordering, the intersection of a class abstraction is less than or equal to any of its members. (Contributed by NM, 3-Jul-2005.) |
Ref | Expression |
---|---|
intmin3.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
intmin3.3 | ⊢ 𝜓 |
Ref | Expression |
---|---|
intmin3 | ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intmin3.3 | . . 3 ⊢ 𝜓 | |
2 | intmin3.2 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | elabg 3612 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
4 | 1, 3 | mpbiri 259 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝑥 ∣ 𝜑}) |
5 | intss1 4901 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴) | |
6 | 4, 5 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2104 {cab 2713 ⊆ wss 3892 ∩ cint 4886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3439 df-in 3899 df-ss 3909 df-int 4887 |
This theorem is referenced by: intabs 5275 intid 5382 |
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