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Mirrors > Home > MPE Home > Th. List > uniprgOLD | Structured version Visualization version GIF version |
Description: Obsolete version of unipr 4870 as of 1-Sep-2024. (Contributed by NM, 25-Aug-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
uniprgOLD | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1 4681 | . . . 4 ⊢ (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦}) | |
2 | 1 | unieqd 4866 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ {𝑥, 𝑦} = ∪ {𝐴, 𝑦}) |
3 | uneq1 4103 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦)) | |
4 | 2, 3 | eqeq12d 2752 | . 2 ⊢ (𝑥 = 𝐴 → (∪ {𝑥, 𝑦} = (𝑥 ∪ 𝑦) ↔ ∪ {𝐴, 𝑦} = (𝐴 ∪ 𝑦))) |
5 | preq2 4682 | . . . 4 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵}) | |
6 | 5 | unieqd 4866 | . . 3 ⊢ (𝑦 = 𝐵 → ∪ {𝐴, 𝑦} = ∪ {𝐴, 𝐵}) |
7 | uneq2 4104 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵)) | |
8 | 6, 7 | eqeq12d 2752 | . 2 ⊢ (𝑦 = 𝐵 → (∪ {𝐴, 𝑦} = (𝐴 ∪ 𝑦) ↔ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))) |
9 | vex 3445 | . . 3 ⊢ 𝑥 ∈ V | |
10 | vex 3445 | . . 3 ⊢ 𝑦 ∈ V | |
11 | 9, 10 | unipr 4870 | . 2 ⊢ ∪ {𝑥, 𝑦} = (𝑥 ∪ 𝑦) |
12 | 4, 8, 11 | vtocl2g 3519 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∪ cun 3896 {cpr 4575 ∪ cuni 4852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3443 df-un 3903 df-in 3905 df-ss 3915 df-sn 4574 df-pr 4576 df-uni 4853 |
This theorem is referenced by: (None) |
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