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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 4999 | . 2 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴} | |
2 | vex 3482 | . . . 4 ⊢ 𝑦 ∈ V | |
3 | ral0 4519 | . . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴 | |
4 | 2, 3 | 2th 264 | . . 3 ⊢ (𝑦 ∈ V ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴) |
5 | 4 | eqabi 2875 | . 2 ⊢ V = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴} |
6 | 1, 5 | eqtr4i 2766 | 1 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 {cab 2712 ∀wral 3059 Vcvv 3478 ∅c0 4339 ∩ ciin 4997 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-v 3480 df-dif 3966 df-nul 4340 df-iin 4999 |
This theorem is referenced by: iinrab2 5075 iinvdif 5085 riin0 5087 iin0 5368 xpriindi 5850 cmpfi 23432 ptbasfi 23605 pol0N 39892 |
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