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Mirrors > Home > MPE Home > Th. List > intid | Structured version Visualization version GIF version |
Description: The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.) |
Ref | Expression |
---|---|
intid.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
intid | ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5323 | . . 3 ⊢ {𝐴} ∈ V | |
2 | eleq2 2901 | . . . 4 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴})) | |
3 | intid.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
4 | 3 | snid 4594 | . . . 4 ⊢ 𝐴 ∈ {𝐴} |
5 | 2, 4 | intmin3 4896 | . . 3 ⊢ ({𝐴} ∈ V → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴}) |
6 | 1, 5 | ax-mp 5 | . 2 ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴} |
7 | 3 | elintab 4879 | . . . 4 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥(𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)) |
8 | id 22 | . . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥) | |
9 | 7, 8 | mpgbir 1796 | . . 3 ⊢ 𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} |
10 | snssi 4734 | . . 3 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} → {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥}) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} |
12 | 6, 11 | eqssi 3982 | 1 ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {cab 2799 Vcvv 3494 ⊆ wss 3935 {csn 4560 ∩ 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 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 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-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-sn 4561 df-pr 4563 df-int 4869 |
This theorem is referenced by: (None) |
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