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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 4994 | . 2 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴} | |
| 2 | vex 3484 | . . . 4 ⊢ 𝑦 ∈ V | |
| 3 | ral0 4513 | . . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴 | |
| 4 | 2, 3 | 2th 264 | . . 3 ⊢ (𝑦 ∈ V ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴) | 
| 5 | 4 | eqabi 2877 | . 2 ⊢ V = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴} | 
| 6 | 1, 5 | eqtr4i 2768 | 1 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = V | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 Vcvv 3480 ∅c0 4333 ∩ ciin 4992 | 
| 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-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-v 3482 df-dif 3954 df-nul 4334 df-iin 4994 | 
| This theorem is referenced by: iinrab2 5070 iinvdif 5080 riin0 5082 iin0 5362 xpriindi 5847 cmpfi 23416 ptbasfi 23589 pol0N 39911 | 
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