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| Mirrors > Home > MPE Home > Th. List > eusv4 | Structured version Visualization version GIF version | ||
| Description: Two ways to express single-valuedness of a class expression 𝐵(𝑦). (Contributed by NM, 27-Oct-2010.) | 
| Ref | Expression | 
|---|---|
| eusv4.1 | ⊢ 𝐵 ∈ V | 
| Ref | Expression | 
|---|---|
| eusv4 | ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | reusv2lem3 5399 | . 2 ⊢ (∀𝑦 ∈ 𝐴 𝐵 ∈ V → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)) | |
| 2 | eusv4.1 | . . 3 ⊢ 𝐵 ∈ V | |
| 3 | 2 | a1i 11 | . 2 ⊢ (𝑦 ∈ 𝐴 → 𝐵 ∈ V) | 
| 4 | 1, 3 | mprg 3066 | 1 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1539 ∈ wcel 2107 ∃!weu 2567 ∀wral 3060 ∃wrex 3069 Vcvv 3479 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-nul 5305 ax-pow 5364 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-v 3481 df-dif 3953 df-nul 4333 | 
| This theorem is referenced by: (None) | 
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