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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 2981 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦) | |
2 | id 22 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴) | |
3 | 1, 2 | nfeqd 2991 | . . 3 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴) |
4 | 3 | alrimiv 1927 | . 2 ⊢ (Ⅎ𝑥𝐴 → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) |
5 | df-nfc 2966 | . . . . 5 ⊢ (Ⅎ𝑥{𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴}) | |
6 | velsn 4586 | . . . . . . 7 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴) | |
7 | 6 | nfbii 1851 | . . . . . 6 ⊢ (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴) |
8 | 7 | albii 1819 | . . . . 5 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) |
9 | 5, 8 | sylbbr 238 | . . . 4 ⊢ (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥{𝐴}) |
10 | 9 | nfunid 4847 | . . 3 ⊢ (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥∪ {𝐴}) |
11 | nfa1 2154 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 𝐴 ∈ 𝑉 | |
12 | unisng 4860 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) | |
13 | 12 | sps 2183 | . . . 4 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) |
14 | 11, 13 | nfceqdf 2975 | . . 3 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥∪ {𝐴} ↔ Ⅎ𝑥𝐴)) |
15 | 10, 14 | syl5ib 246 | . 2 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)) |
16 | 4, 15 | impbid2 228 | 1 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1534 = wceq 1536 Ⅎwnf 1783 ∈ wcel 2113 Ⅎwnfc 2964 {csn 4570 ∪ cuni 4841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-v 3499 df-un 3944 df-in 3946 df-ss 3955 df-sn 4571 df-pr 4573 df-uni 4842 |
This theorem is referenced by: eusv2nf 5299 |
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