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Mirrors > Home > MPE Home > Th. List > eusv2nf | Structured version Visualization version GIF version |
Description: Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by Mario Carneiro, 18-Nov-2016.) |
Ref | Expression |
---|---|
eusv2.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
eusv2nf | ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfeu1 2649 | . . . 4 ⊢ Ⅎ𝑦∃!𝑦∃𝑥 𝑦 = 𝐴 | |
2 | nfe1 2151 | . . . . . . 7 ⊢ Ⅎ𝑥∃𝑥 𝑦 = 𝐴 | |
3 | 2 | nfeuw 2654 | . . . . . 6 ⊢ Ⅎ𝑥∃!𝑦∃𝑥 𝑦 = 𝐴 |
4 | eusv2.1 | . . . . . . . . 9 ⊢ 𝐴 ∈ V | |
5 | 4 | isseti 3455 | . . . . . . . 8 ⊢ ∃𝑦 𝑦 = 𝐴 |
6 | 19.8a 2178 | . . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ∃𝑥 𝑦 = 𝐴) | |
7 | 6 | ancri 553 | . . . . . . . 8 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 ∧ 𝑦 = 𝐴)) |
8 | 5, 7 | eximii 1838 | . . . . . . 7 ⊢ ∃𝑦(∃𝑥 𝑦 = 𝐴 ∧ 𝑦 = 𝐴) |
9 | eupick 2695 | . . . . . . 7 ⊢ ((∃!𝑦∃𝑥 𝑦 = 𝐴 ∧ ∃𝑦(∃𝑥 𝑦 = 𝐴 ∧ 𝑦 = 𝐴)) → (∃𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴)) | |
10 | 8, 9 | mpan2 690 | . . . . . 6 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴)) |
11 | 3, 10 | alrimi 2211 | . . . . 5 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → ∀𝑥(∃𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴)) |
12 | nf6 2287 | . . . . 5 ⊢ (Ⅎ𝑥 𝑦 = 𝐴 ↔ ∀𝑥(∃𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴)) | |
13 | 11, 12 | sylibr 237 | . . . 4 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → Ⅎ𝑥 𝑦 = 𝐴) |
14 | 1, 13 | alrimi 2211 | . . 3 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) |
15 | dfnfc2 4822 | . . . 4 ⊢ (∀𝑥 𝐴 ∈ V → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)) | |
16 | 15, 4 | mpg 1799 | . . 3 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) |
17 | 14, 16 | sylibr 237 | . 2 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴) |
18 | eusvnfb 5259 | . . . 4 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V)) | |
19 | 4, 18 | mpbiran2 709 | . . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴) |
20 | eusv2i 5260 | . . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → ∃!𝑦∃𝑥 𝑦 = 𝐴) | |
21 | 19, 20 | sylbir 238 | . 2 ⊢ (Ⅎ𝑥𝐴 → ∃!𝑦∃𝑥 𝑦 = 𝐴) |
22 | 17, 21 | impbii 212 | 1 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 = wceq 1538 ∃wex 1781 Ⅎwnf 1785 ∈ wcel 2111 ∃!weu 2628 Ⅎwnfc 2936 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 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-fal 1551 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-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-uni 4801 |
This theorem is referenced by: eusv2 5262 |
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