![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4884 | . 2 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴} | |
2 | vex 3444 | . . . 4 ⊢ 𝑦 ∈ V | |
3 | ral0 4414 | . . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴 | |
4 | 2, 3 | 2th 267 | . . 3 ⊢ (𝑦 ∈ V ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴) |
5 | 4 | abbi2i 2929 | . 2 ⊢ V = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴} |
6 | 1, 5 | eqtr4i 2824 | 1 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 {cab 2776 ∀wral 3106 Vcvv 3441 ∅c0 4243 ∩ ciin 4882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-v 3443 df-dif 3884 df-nul 4244 df-iin 4884 |
This theorem is referenced by: iinrab2 4955 iinvdif 4965 riin0 4967 iin0 5226 xpriindi 5671 cmpfi 22013 ptbasfi 22186 pol0N 37205 |
Copyright terms: Public domain | W3C validator |