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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 5397 | . 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 3067 | 1 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 ∃!weu 2562 ∀wral 3061 ∃wrex 3070 Vcvv 3474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-nul 5305 ax-pow 5362 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-v 3476 df-dif 3950 df-nul 4322 |
This theorem is referenced by: (None) |
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