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Mirrors > Home > ILE Home > Th. List > elvvuni | GIF version |
Description: An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.) |
Ref | Expression |
---|---|
elvvuni | ⊢ (A ∈ (V × V) → ∪ A ∈ A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elvv 4345 | . 2 ⊢ (A ∈ (V × V) ↔ ∃x∃y A = 〈x, y〉) | |
2 | vex 2554 | . . . . . 6 ⊢ x ∈ V | |
3 | vex 2554 | . . . . . 6 ⊢ y ∈ V | |
4 | 2, 3 | uniop 3983 | . . . . 5 ⊢ ∪ 〈x, y〉 = {x, y} |
5 | 2, 3 | opi2 3961 | . . . . 5 ⊢ {x, y} ∈ 〈x, y〉 |
6 | 4, 5 | eqeltri 2107 | . . . 4 ⊢ ∪ 〈x, y〉 ∈ 〈x, y〉 |
7 | unieq 3580 | . . . . 5 ⊢ (A = 〈x, y〉 → ∪ A = ∪ 〈x, y〉) | |
8 | id 19 | . . . . 5 ⊢ (A = 〈x, y〉 → A = 〈x, y〉) | |
9 | 7, 8 | eleq12d 2105 | . . . 4 ⊢ (A = 〈x, y〉 → (∪ A ∈ A ↔ ∪ 〈x, y〉 ∈ 〈x, y〉)) |
10 | 6, 9 | mpbiri 157 | . . 3 ⊢ (A = 〈x, y〉 → ∪ A ∈ A) |
11 | 10 | exlimivv 1773 | . 2 ⊢ (∃x∃y A = 〈x, y〉 → ∪ A ∈ A) |
12 | 1, 11 | sylbi 114 | 1 ⊢ (A ∈ (V × V) → ∪ A ∈ A) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∃wex 1378 ∈ wcel 1390 Vcvv 2551 {cpr 3368 〈cop 3370 ∪ cuni 3571 × 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-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-uni 3572 df-opab 3810 df-xp 4294 |
This theorem is referenced by: unielxp 5742 |
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