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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 3470 | . . 3 ⊢ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = {𝑥 ∣ 𝐴 ⊆ 𝑥} | |
2 | 1 | inteqi 4842 | . 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} |
3 | intmin2.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | intmin 4858 | . . 3 ⊢ (𝐴 ∈ V → ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴 |
6 | 2, 5 | eqtr3i 2823 | 1 ⊢ ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 {cab 2776 {crab 3110 Vcvv 3441 ⊆ 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-ral 3111 df-rab 3115 df-v 3443 df-in 3888 df-ss 3898 df-int 4839 |
This theorem is referenced by: dfid7 40312 |
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