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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 3523 | . . 3 ⊢ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = {𝑥 ∣ 𝐴 ⊆ 𝑥} | |
2 | 1 | inteqi 4872 | . 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} |
3 | intmin2.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | intmin 4888 | . . 3 ⊢ (𝐴 ∈ V → ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴 |
6 | 2, 5 | eqtr3i 2846 | 1 ⊢ ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 {cab 2799 {crab 3142 Vcvv 3494 ⊆ wss 3935 ∩ cint 4868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-in 3942 df-ss 3951 df-int 4869 |
This theorem is referenced by: dfid7 39965 |
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