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Mirrors > Home > MPE Home > Th. List > eusvnfb | Structured version Visualization version 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 5391 | . . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴) | |
2 | euex 2572 | . . . 4 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → ∃𝑦∀𝑥 𝑦 = 𝐴) | |
3 | eqvisset 3492 | . . . . . 6 ⊢ (𝑦 = 𝐴 → 𝐴 ∈ V) | |
4 | 3 | sps 2179 | . . . . 5 ⊢ (∀𝑥 𝑦 = 𝐴 → 𝐴 ∈ V) |
5 | 4 | exlimiv 1934 | . . . 4 ⊢ (∃𝑦∀𝑥 𝑦 = 𝐴 → 𝐴 ∈ V) |
6 | 2, 5 | syl 17 | . . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → 𝐴 ∈ V) |
7 | 1, 6 | jca 513 | . 2 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → (Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V)) |
8 | isset 3488 | . . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴) | |
9 | nfcvd 2905 | . . . . . . . 8 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦) | |
10 | id 22 | . . . . . . . 8 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴) | |
11 | 9, 10 | nfeqd 2914 | . . . . . . 7 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴) |
12 | 11 | nf5rd 2190 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 → (𝑦 = 𝐴 → ∀𝑥 𝑦 = 𝐴)) |
13 | 12 | eximdv 1921 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → (∃𝑦 𝑦 = 𝐴 → ∃𝑦∀𝑥 𝑦 = 𝐴)) |
14 | 8, 13 | biimtrid 241 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ V → ∃𝑦∀𝑥 𝑦 = 𝐴)) |
15 | 14 | imp 408 | . . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V) → ∃𝑦∀𝑥 𝑦 = 𝐴) |
16 | eusv1 5390 | . . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ ∃𝑦∀𝑥 𝑦 = 𝐴) | |
17 | 15, 16 | sylibr 233 | . 2 ⊢ ((Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V) → ∃!𝑦∀𝑥 𝑦 = 𝐴) |
18 | 7, 17 | impbii 208 | 1 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∀wal 1540 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∃!weu 2563 Ⅎwnfc 2884 Vcvv 3475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-nul 4324 |
This theorem is referenced by: eusv2nf 5394 eusv2 5395 |
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