Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > int0 | 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.) |
Ref | Expression |
---|---|
int0 | ⊢ ∩ ∅ = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3255 | . . . . . 6 ⊢ ¬ 𝑦 ∈ ∅ | |
2 | 1 | pm2.21i 607 | . . . . 5 ⊢ (𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) |
3 | 2 | ax-gen 1378 | . . . 4 ⊢ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) |
4 | equid 1629 | . . . 4 ⊢ 𝑥 = 𝑥 | |
5 | 3, 4 | 2th 172 | . . 3 ⊢ (∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) ↔ 𝑥 = 𝑥) |
6 | 5 | abbii 2194 | . 2 ⊢ {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)} = {𝑥 ∣ 𝑥 = 𝑥} |
7 | df-int 3637 | . 2 ⊢ ∩ ∅ = {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)} | |
8 | df-v 2603 | . 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
9 | 6, 7, 8 | 3eqtr4i 2111 | 1 ⊢ ∩ ∅ = V |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1282 = wceq 1284 ∈ wcel 1433 {cab 2067 Vcvv 2601 ∅c0 3251 ∩ cint 3636 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-dif 2975 df-nul 3252 df-int 3637 |
This theorem is referenced by: rint0 3675 intexr 3925 bj-intexr 10699 |
Copyright terms: Public domain | W3C validator |