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| Mirrors > Home > MPE Home > Th. List > intmin2 | Structured version Visualization version GIF version | ||
| Description: Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.) |
| Ref | Expression |
|---|---|
| intmin2.1 | ⊢ 𝐴 ∈ V |
| Ref | Expression |
|---|---|
| intmin2 | ⊢ ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} = 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rabab 3512 | . . 3 ⊢ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = {𝑥 ∣ 𝐴 ⊆ 𝑥} | |
| 2 | 1 | inteqi 4950 | . 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} |
| 3 | intmin2.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 4 | intmin 4968 | . . 3 ⊢ (𝐴 ∈ V → ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴) | |
| 5 | 3, 4 | ax-mp 5 | . 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴 |
| 6 | 2, 5 | eqtr3i 2767 | 1 ⊢ ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} = 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 {cab 2714 {crab 3436 Vcvv 3480 ⊆ wss 3951 ∩ cint 4946 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-ss 3968 df-int 4947 |
| This theorem is referenced by: dfid7 43625 |
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