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Mirrors > Home > ILE Home > Th. List > xpmlem | GIF version |
Description: The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 11-Dec-2018.) |
Ref | Expression |
---|---|
xpmlem | ⊢ ((∃x x ∈ A ∧ ∃y y ∈ B) ↔ ∃z z ∈ (A × B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eeanv 1804 | . . 3 ⊢ (∃x∃y(x ∈ A ∧ y ∈ B) ↔ (∃x x ∈ A ∧ ∃y y ∈ B)) | |
2 | vex 2554 | . . . . . 6 ⊢ x ∈ V | |
3 | vex 2554 | . . . . . 6 ⊢ y ∈ V | |
4 | 2, 3 | opex 3957 | . . . . 5 ⊢ 〈x, y〉 ∈ V |
5 | eleq1 2097 | . . . . . 6 ⊢ (z = 〈x, y〉 → (z ∈ (A × B) ↔ 〈x, y〉 ∈ (A × B))) | |
6 | opelxp 4317 | . . . . . 6 ⊢ (〈x, y〉 ∈ (A × B) ↔ (x ∈ A ∧ y ∈ B)) | |
7 | 5, 6 | syl6bb 185 | . . . . 5 ⊢ (z = 〈x, y〉 → (z ∈ (A × B) ↔ (x ∈ A ∧ y ∈ B))) |
8 | 4, 7 | spcev 2641 | . . . 4 ⊢ ((x ∈ A ∧ y ∈ B) → ∃z z ∈ (A × B)) |
9 | 8 | exlimivv 1773 | . . 3 ⊢ (∃x∃y(x ∈ A ∧ y ∈ B) → ∃z z ∈ (A × B)) |
10 | 1, 9 | sylbir 125 | . 2 ⊢ ((∃x x ∈ A ∧ ∃y y ∈ B) → ∃z z ∈ (A × B)) |
11 | elxp 4305 | . . . . 5 ⊢ (z ∈ (A × B) ↔ ∃x∃y(z = 〈x, y〉 ∧ (x ∈ A ∧ y ∈ B))) | |
12 | simpr 103 | . . . . . 6 ⊢ ((z = 〈x, y〉 ∧ (x ∈ A ∧ y ∈ B)) → (x ∈ A ∧ y ∈ B)) | |
13 | 12 | 2eximi 1489 | . . . . 5 ⊢ (∃x∃y(z = 〈x, y〉 ∧ (x ∈ A ∧ y ∈ B)) → ∃x∃y(x ∈ A ∧ y ∈ B)) |
14 | 11, 13 | sylbi 114 | . . . 4 ⊢ (z ∈ (A × B) → ∃x∃y(x ∈ A ∧ y ∈ B)) |
15 | 14 | exlimiv 1486 | . . 3 ⊢ (∃z z ∈ (A × B) → ∃x∃y(x ∈ A ∧ y ∈ B)) |
16 | 15, 1 | sylib 127 | . 2 ⊢ (∃z z ∈ (A × B) → (∃x x ∈ A ∧ ∃y y ∈ B)) |
17 | 10, 16 | impbii 117 | 1 ⊢ ((∃x x ∈ A ∧ ∃y y ∈ B) ↔ ∃z z ∈ (A × B)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1242 ∃wex 1378 ∈ wcel 1390 〈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-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-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-opab 3810 df-xp 4294 |
This theorem is referenced by: xpm 4688 |
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