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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 3314 | . . . . . 6 ⊢ ¬ 𝑦 ∈ ∅ | |
2 | 1 | pm2.21i 615 | . . . . 5 ⊢ (𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) |
3 | 2 | ax-gen 1393 | . . . 4 ⊢ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) |
4 | equid 1645 | . . . 4 ⊢ 𝑥 = 𝑥 | |
5 | 3, 4 | 2th 173 | . . 3 ⊢ (∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) ↔ 𝑥 = 𝑥) |
6 | 5 | abbii 2215 | . 2 ⊢ {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)} = {𝑥 ∣ 𝑥 = 𝑥} |
7 | df-int 3719 | . 2 ⊢ ∩ ∅ = {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)} | |
8 | df-v 2643 | . 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
9 | 6, 7, 8 | 3eqtr4i 2130 | 1 ⊢ ∩ ∅ = V |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1297 = wceq 1299 ∈ wcel 1448 {cab 2086 Vcvv 2641 ∅c0 3310 ∩ cint 3718 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 df-dif 3023 df-nul 3311 df-int 3719 |
This theorem is referenced by: rint0 3757 intexr 4015 fiintim 6746 bj-intexr 12687 |
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