Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > elvvuni | Structured version Visualization version 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 5661 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
2 | vex 3436 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3436 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | uniop 5429 | . . . . 5 ⊢ ∪ 〈𝑥, 𝑦〉 = {𝑥, 𝑦} |
5 | 2, 3 | opi2 5384 | . . . . 5 ⊢ {𝑥, 𝑦} ∈ 〈𝑥, 𝑦〉 |
6 | 4, 5 | eqeltri 2835 | . . . 4 ⊢ ∪ 〈𝑥, 𝑦〉 ∈ 〈𝑥, 𝑦〉 |
7 | unieq 4850 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∪ 𝐴 = ∪ 〈𝑥, 𝑦〉) | |
8 | id 22 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝐴 = 〈𝑥, 𝑦〉) | |
9 | 7, 8 | eleq12d 2833 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (∪ 𝐴 ∈ 𝐴 ↔ ∪ 〈𝑥, 𝑦〉 ∈ 〈𝑥, 𝑦〉)) |
10 | 6, 9 | mpbiri 257 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∪ 𝐴 ∈ 𝐴) |
11 | 10 | exlimivv 1935 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 → ∪ 𝐴 ∈ 𝐴) |
12 | 1, 11 | sylbi 216 | 1 ⊢ (𝐴 ∈ (V × V) → ∪ 𝐴 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∃wex 1782 ∈ wcel 2106 Vcvv 3432 {cpr 4563 〈cop 4567 ∪ cuni 4839 × cxp 5587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-opab 5137 df-xp 5595 |
This theorem is referenced by: unielxp 7869 |
Copyright terms: Public domain | W3C validator |