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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 4458 | . 2 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴} | |
2 | vex 3176 | . . . 4 ⊢ 𝑦 ∈ V | |
3 | ral0 4028 | . . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴 | |
4 | 2, 3 | 2th 253 | . . 3 ⊢ (𝑦 ∈ V ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴) |
5 | 4 | abbi2i 2725 | . 2 ⊢ V = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴} |
6 | 1, 5 | eqtr4i 2635 | 1 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 {cab 2596 ∀wral 2896 Vcvv 3173 ∅c0 3874 ∩ ciin 4456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-dif 3543 df-nul 3875 df-iin 4458 |
This theorem is referenced by: iinrab2 4519 iinvdif 4528 riin0 4530 iin0 4765 xpriindi 5180 cmpfi 21021 ptbasfi 21194 pol0N 34213 |
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