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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 4706 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
2 | vex 2755 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 2755 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | uniop 4273 | . . . . 5 ⊢ ∪ 〈𝑥, 𝑦〉 = {𝑥, 𝑦} |
5 | 2, 3 | opi2 4251 | . . . . 5 ⊢ {𝑥, 𝑦} ∈ 〈𝑥, 𝑦〉 |
6 | 4, 5 | eqeltri 2262 | . . . 4 ⊢ ∪ 〈𝑥, 𝑦〉 ∈ 〈𝑥, 𝑦〉 |
7 | unieq 3833 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∪ 𝐴 = ∪ 〈𝑥, 𝑦〉) | |
8 | id 19 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝐴 = 〈𝑥, 𝑦〉) | |
9 | 7, 8 | eleq12d 2260 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (∪ 𝐴 ∈ 𝐴 ↔ ∪ 〈𝑥, 𝑦〉 ∈ 〈𝑥, 𝑦〉)) |
10 | 6, 9 | mpbiri 168 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∪ 𝐴 ∈ 𝐴) |
11 | 10 | exlimivv 1908 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 → ∪ 𝐴 ∈ 𝐴) |
12 | 1, 11 | sylbi 121 | 1 ⊢ (𝐴 ∈ (V × V) → ∪ 𝐴 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∃wex 1503 ∈ wcel 2160 Vcvv 2752 {cpr 3608 〈cop 3610 ∪ cuni 3824 × cxp 4642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-opab 4080 df-xp 4650 |
This theorem is referenced by: unielxp 6200 |
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