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Mirrors > Home > MPE Home > Th. List > int0 | Structured version Visualization version GIF version |
Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) (Proof shortened by JJ, 26-Jul-2021.) |
Ref | Expression |
---|---|
int0 | ⊢ ∩ ∅ = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ral0 4028 | . . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥 | |
2 | vex 3176 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 2 | elint2 4417 | . . . 4 ⊢ (𝑦 ∈ ∩ ∅ ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥) |
4 | 1, 3 | mpbir 220 | . . 3 ⊢ 𝑦 ∈ ∩ ∅ |
5 | 4, 2 | 2th 253 | . 2 ⊢ (𝑦 ∈ ∩ ∅ ↔ 𝑦 ∈ V) |
6 | 5 | eqriv 2607 | 1 ⊢ ∩ ∅ = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∅c0 3874 ∩ cint 4410 |
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-int 4411 |
This theorem is referenced by: unissint 4436 uniintsn 4449 rint0 4452 intex 4747 intnex 4748 oev2 7490 fiint 8122 elfi2 8203 fi0 8209 cardmin2 8707 00lsp 18802 cmpfi 21021 ptbasfi 21194 fbssint 21452 fclscmp 21644 rankeq1o 31448 heibor1lem 32778 |
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