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Mirrors > Home > MPE Home > Th. List > 0iin | Structured version Visualization version GIF version |
Description: An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.) |
Ref | Expression |
---|---|
0iin | ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iin 4713 | . 2 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴} | |
2 | vex 3388 | . . . 4 ⊢ 𝑦 ∈ V | |
3 | ral0 4269 | . . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴 | |
4 | 2, 3 | 2th 256 | . . 3 ⊢ (𝑦 ∈ V ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴) |
5 | 4 | abbi2i 2915 | . 2 ⊢ V = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴} |
6 | 1, 5 | eqtr4i 2824 | 1 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 {cab 2785 ∀wral 3089 Vcvv 3385 ∅c0 4115 ∩ ciin 4711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-v 3387 df-dif 3772 df-nul 4116 df-iin 4713 |
This theorem is referenced by: iinrab2 4773 iinvdif 4782 riin0 4784 iin0 5031 xpriindi 5462 cmpfi 21540 ptbasfi 21713 pol0N 35930 |
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