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Mirrors > Home > ILE Home > Th. List > eusvnfb | GIF version |
Description: Two ways to say that A(x) is a set expression that does not depend on x. (Contributed by Mario Carneiro, 18-Nov-2016.) |
Ref | Expression |
---|---|
eusvnfb | ⊢ (∃!y∀x y = A ↔ (ℲxA ∧ A ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eusvnf 4151 | . . 3 ⊢ (∃!y∀x y = A → ℲxA) | |
2 | euex 1927 | . . . 4 ⊢ (∃!y∀x y = A → ∃y∀x y = A) | |
3 | id 19 | . . . . . . 7 ⊢ (y = A → y = A) | |
4 | vex 2554 | . . . . . . 7 ⊢ y ∈ V | |
5 | 3, 4 | syl6eqelr 2126 | . . . . . 6 ⊢ (y = A → A ∈ V) |
6 | 5 | sps 1427 | . . . . 5 ⊢ (∀x y = A → A ∈ V) |
7 | 6 | exlimiv 1486 | . . . 4 ⊢ (∃y∀x y = A → A ∈ V) |
8 | 2, 7 | syl 14 | . . 3 ⊢ (∃!y∀x y = A → A ∈ V) |
9 | 1, 8 | jca 290 | . 2 ⊢ (∃!y∀x y = A → (ℲxA ∧ A ∈ V)) |
10 | isset 2555 | . . . . 5 ⊢ (A ∈ V ↔ ∃y y = A) | |
11 | nfcvd 2176 | . . . . . . . 8 ⊢ (ℲxA → Ⅎxy) | |
12 | id 19 | . . . . . . . 8 ⊢ (ℲxA → ℲxA) | |
13 | 11, 12 | nfeqd 2189 | . . . . . . 7 ⊢ (ℲxA → Ⅎx y = A) |
14 | 13 | nfrd 1410 | . . . . . 6 ⊢ (ℲxA → (y = A → ∀x y = A)) |
15 | 14 | eximdv 1757 | . . . . 5 ⊢ (ℲxA → (∃y y = A → ∃y∀x y = A)) |
16 | 10, 15 | syl5bi 141 | . . . 4 ⊢ (ℲxA → (A ∈ V → ∃y∀x y = A)) |
17 | 16 | imp 115 | . . 3 ⊢ ((ℲxA ∧ A ∈ V) → ∃y∀x y = A) |
18 | eusv1 4150 | . . 3 ⊢ (∃!y∀x y = A ↔ ∃y∀x y = A) | |
19 | 17, 18 | sylibr 137 | . 2 ⊢ ((ℲxA ∧ A ∈ V) → ∃!y∀x y = A) |
20 | 9, 19 | impbii 117 | 1 ⊢ (∃!y∀x y = A ↔ (ℲxA ∧ A ∈ V)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∀wal 1240 = wceq 1242 ∃wex 1378 ∈ wcel 1390 ∃!weu 1897 Ⅎwnfc 2162 Vcvv 2551 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-sbc 2759 df-csb 2847 |
This theorem is referenced by: eusv2nf 4154 eusv2 4155 |
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