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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 3614 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
4 | 1, 3 | mpbiri 261 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝑥 ∣ 𝜑}) |
5 | intss1 4853 | . 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 209 = wceq 1538 ∈ wcel 2111 {cab 2776 ⊆ wss 3881 ∩ cint 4838 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-in 3888 df-ss 3898 df-int 4839 |
This theorem is referenced by: intabs 5209 intid 5315 |
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