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Mirrors > Home > MPE Home > Th. List > intminss | Structured version Visualization version GIF version |
Description: Under subset ordering, the intersection of a restricted class abstraction is less than or equal to any of its members. (Contributed by NM, 7-Sep-2013.) |
Ref | Expression |
---|---|
intminss.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
intminss | ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intminss.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
2 | 1 | elrab 3331 | . 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
3 | intss1 4427 | . 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} → ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐴) | |
4 | 2, 3 | sylbir 224 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 ⊆ 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-rab 2905 df-v 3175 df-in 3547 df-ss 3554 df-int 4411 |
This theorem is referenced by: onintss 5692 knatar 6507 cardonle 8666 coftr 8978 wuncss 9446 ist1-3 20963 sigagenss 29539 ldgenpisyslem1 29553 dynkin 29557 nodenselem5 31084 nobndlem6 31096 nobndlem8 31098 fneint 31513 igenmin 33033 pclclN 34195 dfrcl2 36985 |
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