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Mirrors > Home > MPE Home > Th. List > dfnfc2 | Structured version Visualization version GIF version |
Description: An alternative statement of the effective freeness of a class 𝐴, when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.) (Proof shortened by JJ, 26-Jul-2021.) |
Ref | Expression |
---|---|
dfnfc2 | ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcvd 2908 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦) | |
2 | id 22 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴) | |
3 | 1, 2 | nfeqd 2917 | . . 3 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴) |
4 | 3 | alrimiv 1930 | . 2 ⊢ (Ⅎ𝑥𝐴 → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) |
5 | df-nfc 2889 | . . . . 5 ⊢ (Ⅎ𝑥{𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴}) | |
6 | velsn 4577 | . . . . . . 7 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴) | |
7 | 6 | nfbii 1854 | . . . . . 6 ⊢ (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴) |
8 | 7 | albii 1822 | . . . . 5 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) |
9 | 5, 8 | sylbbr 235 | . . . 4 ⊢ (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥{𝐴}) |
10 | 9 | nfunid 4845 | . . 3 ⊢ (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥∪ {𝐴}) |
11 | nfa1 2148 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 𝐴 ∈ 𝑉 | |
12 | unisng 4860 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) | |
13 | 12 | sps 2178 | . . . 4 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) |
14 | 11, 13 | nfceqdf 2902 | . . 3 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥∪ {𝐴} ↔ Ⅎ𝑥𝐴)) |
15 | 10, 14 | syl5ib 243 | . 2 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)) |
16 | 4, 15 | impbid2 225 | 1 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1537 = wceq 1539 Ⅎwnf 1786 ∈ wcel 2106 Ⅎwnfc 2887 {csn 4561 ∪ cuni 4839 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-v 3434 df-un 3892 df-in 3894 df-ss 3904 df-sn 4562 df-pr 4564 df-uni 4840 |
This theorem is referenced by: eusv2nf 5318 |
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