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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 3367 | . . . . . 6 ⊢ ¬ 𝑦 ∈ ∅ | |
2 | 1 | pm2.21i 635 | . . . . 5 ⊢ (𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) |
3 | 2 | ax-gen 1425 | . . . 4 ⊢ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) |
4 | equid 1677 | . . . 4 ⊢ 𝑥 = 𝑥 | |
5 | 3, 4 | 2th 173 | . . 3 ⊢ (∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) ↔ 𝑥 = 𝑥) |
6 | 5 | abbii 2255 | . 2 ⊢ {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)} = {𝑥 ∣ 𝑥 = 𝑥} |
7 | df-int 3772 | . 2 ⊢ ∩ ∅ = {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)} | |
8 | df-v 2688 | . 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
9 | 6, 7, 8 | 3eqtr4i 2170 | 1 ⊢ ∩ ∅ = V |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1329 = wceq 1331 ∈ wcel 1480 {cab 2125 Vcvv 2686 ∅c0 3363 ∩ cint 3771 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-dif 3073 df-nul 3364 df-int 3772 |
This theorem is referenced by: rint0 3810 intexr 4075 fiintim 6817 elfi2 6860 fi0 6863 bj-intexr 13106 |
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