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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 3372 | . . . . . 6 ⊢ ¬ 𝑦 ∈ ∅ | |
2 | 1 | pm2.21i 636 | . . . . 5 ⊢ (𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) |
3 | 2 | ax-gen 1426 | . . . 4 ⊢ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) |
4 | equid 1678 | . . . 4 ⊢ 𝑥 = 𝑥 | |
5 | 3, 4 | 2th 173 | . . 3 ⊢ (∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) ↔ 𝑥 = 𝑥) |
6 | 5 | abbii 2256 | . 2 ⊢ {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)} = {𝑥 ∣ 𝑥 = 𝑥} |
7 | df-int 3780 | . 2 ⊢ ∩ ∅ = {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)} | |
8 | df-v 2691 | . 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
9 | 6, 7, 8 | 3eqtr4i 2171 | 1 ⊢ ∩ ∅ = V |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1330 = wceq 1332 ∈ wcel 1481 {cab 2126 Vcvv 2689 ∅c0 3368 ∩ cint 3779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-dif 3078 df-nul 3369 df-int 3780 |
This theorem is referenced by: rint0 3818 intexr 4083 fiintim 6825 elfi2 6868 fi0 6871 bj-intexr 13277 |
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