Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2980 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦) | |
2 | id 22 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴) | |
3 | 1, 2 | nfeqd 2990 | . . 3 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴) |
4 | 3 | alrimiv 1928 | . 2 ⊢ (Ⅎ𝑥𝐴 → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) |
5 | df-nfc 2965 | . . . . 5 ⊢ (Ⅎ𝑥{𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴}) | |
6 | velsn 4585 | . . . . . . 7 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴) | |
7 | 6 | nfbii 1852 | . . . . . 6 ⊢ (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴) |
8 | 7 | albii 1820 | . . . . 5 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) |
9 | 5, 8 | sylbbr 238 | . . . 4 ⊢ (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥{𝐴}) |
10 | 9 | nfunid 4846 | . . 3 ⊢ (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥∪ {𝐴}) |
11 | nfa1 2155 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 𝐴 ∈ 𝑉 | |
12 | unisng 4859 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) | |
13 | 12 | sps 2184 | . . . 4 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) |
14 | 11, 13 | nfceqdf 2974 | . . 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 1535 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2963 {csn 4569 ∪ cuni 4840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-v 3498 df-un 3943 df-in 3945 df-ss 3954 df-sn 4570 df-pr 4572 df-uni 4841 |
This theorem is referenced by: eusv2nf 5298 |
Copyright terms: Public domain | W3C validator |