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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 5711 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩) | |
2 | vex 3452 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3452 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | uniop 5477 | . . . . 5 ⊢ ∪ ⟨𝑥, 𝑦⟩ = {𝑥, 𝑦} |
5 | 2, 3 | opi2 5431 | . . . . 5 ⊢ {𝑥, 𝑦} ∈ ⟨𝑥, 𝑦⟩ |
6 | 4, 5 | eqeltri 2834 | . . . 4 ⊢ ∪ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦⟩ |
7 | unieq 4881 | . . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∪ 𝐴 = ∪ ⟨𝑥, 𝑦⟩) | |
8 | id 22 | . . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = ⟨𝑥, 𝑦⟩) | |
9 | 7, 8 | eleq12d 2832 | . . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (∪ 𝐴 ∈ 𝐴 ↔ ∪ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦⟩)) |
10 | 6, 9 | mpbiri 258 | . . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∪ 𝐴 ∈ 𝐴) |
11 | 10 | exlimivv 1936 | . 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 1542 ∃wex 1782 ∈ wcel 2107 Vcvv 3448 {cpr 4593 ⟨cop 4597 ∪ cuni 4870 × cxp 5636 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-opab 5173 df-xp 5644 |
This theorem is referenced by: unielxp 7964 |
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