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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 5302 | . 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 3076 | 1 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2111 ∃!weu 2568 ∀wral 3062 ∃wrex 3063 Vcvv 3415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-nul 5208 ax-pow 5267 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-ne 2942 df-ral 3067 df-rex 3068 df-v 3417 df-dif 3878 df-nul 4247 |
This theorem is referenced by: (None) |
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