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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 5100 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
2 | vex 3176 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3176 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | uniop 4902 | . . . . 5 ⊢ ∪ 〈𝑥, 𝑦〉 = {𝑥, 𝑦} |
5 | 2, 3 | opi2 4865 | . . . . 5 ⊢ {𝑥, 𝑦} ∈ 〈𝑥, 𝑦〉 |
6 | 4, 5 | eqeltri 2684 | . . . 4 ⊢ ∪ 〈𝑥, 𝑦〉 ∈ 〈𝑥, 𝑦〉 |
7 | unieq 4380 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∪ 𝐴 = ∪ 〈𝑥, 𝑦〉) | |
8 | id 22 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝐴 = 〈𝑥, 𝑦〉) | |
9 | 7, 8 | eleq12d 2682 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (∪ 𝐴 ∈ 𝐴 ↔ ∪ 〈𝑥, 𝑦〉 ∈ 〈𝑥, 𝑦〉)) |
10 | 6, 9 | mpbiri 247 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∪ 𝐴 ∈ 𝐴) |
11 | 10 | exlimivv 1847 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 → ∪ 𝐴 ∈ 𝐴) |
12 | 1, 11 | sylbi 206 | 1 ⊢ (𝐴 ∈ (V × V) → ∪ 𝐴 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 {cpr 4127 〈cop 4131 ∪ cuni 4372 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-opab 4644 df-xp 5044 |
This theorem is referenced by: unielxp 7095 |
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