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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 5266 | . 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 3120 | 1 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∃!weu 2628 ∀wral 3106 ∃wrex 3107 Vcvv 3441 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 ax-pow 5231 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-nul 4244 |
This theorem is referenced by: (None) |
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