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Mirrors > Home > ILE Home > Th. List > 0xp | GIF version |
Description: The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 4-Jul-1994.) |
Ref | Expression |
---|---|
0xp | ⊢ (∅ × A) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp 4305 | . . 3 ⊢ (z ∈ (∅ × A) ↔ ∃x∃y(z = 〈x, y〉 ∧ (x ∈ ∅ ∧ y ∈ A))) | |
2 | noel 3222 | . . . . . . 7 ⊢ ¬ x ∈ ∅ | |
3 | simprl 483 | . . . . . . 7 ⊢ ((z = 〈x, y〉 ∧ (x ∈ ∅ ∧ y ∈ A)) → x ∈ ∅) | |
4 | 2, 3 | mto 587 | . . . . . 6 ⊢ ¬ (z = 〈x, y〉 ∧ (x ∈ ∅ ∧ y ∈ A)) |
5 | 4 | nex 1386 | . . . . 5 ⊢ ¬ ∃y(z = 〈x, y〉 ∧ (x ∈ ∅ ∧ y ∈ A)) |
6 | 5 | nex 1386 | . . . 4 ⊢ ¬ ∃x∃y(z = 〈x, y〉 ∧ (x ∈ ∅ ∧ y ∈ A)) |
7 | noel 3222 | . . . 4 ⊢ ¬ z ∈ ∅ | |
8 | 6, 7 | 2false 616 | . . 3 ⊢ (∃x∃y(z = 〈x, y〉 ∧ (x ∈ ∅ ∧ y ∈ A)) ↔ z ∈ ∅) |
9 | 1, 8 | bitri 173 | . 2 ⊢ (z ∈ (∅ × A) ↔ z ∈ ∅) |
10 | 9 | eqriv 2034 | 1 ⊢ (∅ × A) = ∅ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1242 ∃wex 1378 ∈ wcel 1390 ∅c0 3218 〈cop 3370 × cxp 4286 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-opab 3810 df-xp 4294 |
This theorem is referenced by: res0 4559 xp0 4686 xpeq0r 4689 xpdisj1 4690 xpima1 4710 |
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