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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 4539 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
2 | vex 2644 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 2644 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | uniop 4115 | . . . . 5 ⊢ ∪ 〈𝑥, 𝑦〉 = {𝑥, 𝑦} |
5 | 2, 3 | opi2 4093 | . . . . 5 ⊢ {𝑥, 𝑦} ∈ 〈𝑥, 𝑦〉 |
6 | 4, 5 | eqeltri 2172 | . . . 4 ⊢ ∪ 〈𝑥, 𝑦〉 ∈ 〈𝑥, 𝑦〉 |
7 | unieq 3692 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∪ 𝐴 = ∪ 〈𝑥, 𝑦〉) | |
8 | id 19 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝐴 = 〈𝑥, 𝑦〉) | |
9 | 7, 8 | eleq12d 2170 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (∪ 𝐴 ∈ 𝐴 ↔ ∪ 〈𝑥, 𝑦〉 ∈ 〈𝑥, 𝑦〉)) |
10 | 6, 9 | mpbiri 167 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∪ 𝐴 ∈ 𝐴) |
11 | 10 | exlimivv 1835 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 → ∪ 𝐴 ∈ 𝐴) |
12 | 1, 11 | sylbi 120 | 1 ⊢ (𝐴 ∈ (V × V) → ∪ 𝐴 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1299 ∃wex 1436 ∈ wcel 1448 Vcvv 2641 {cpr 3475 〈cop 3477 ∪ cuni 3683 × cxp 4475 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-rex 2381 df-v 2643 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-opab 3930 df-xp 4483 |
This theorem is referenced by: unielxp 6002 |
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