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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 2906 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦) | |
| 2 | id 22 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴) | |
| 3 | 1, 2 | nfeqd 2916 | . . 3 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴) |
| 4 | 3 | alrimiv 1927 | . 2 ⊢ (Ⅎ𝑥𝐴 → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) |
| 5 | df-nfc 2892 | . . . . 5 ⊢ (Ⅎ𝑥{𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴}) | |
| 6 | velsn 4642 | . . . . . . 7 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴) | |
| 7 | 6 | nfbii 1852 | . . . . . 6 ⊢ (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴) |
| 8 | 7 | albii 1819 | . . . . 5 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) |
| 9 | 5, 8 | sylbbr 236 | . . . 4 ⊢ (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥{𝐴}) |
| 10 | 9 | nfunid 4913 | . . 3 ⊢ (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥∪ {𝐴}) |
| 11 | nfa1 2151 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 𝐴 ∈ 𝑉 | |
| 12 | unisng 4925 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) | |
| 13 | 12 | sps 2185 | . . . 4 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) |
| 14 | 11, 13 | nfceqdf 2901 | . . 3 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥∪ {𝐴} ↔ Ⅎ𝑥𝐴)) |
| 15 | 10, 14 | imbitrid 244 | . 2 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)) |
| 16 | 4, 15 | impbid2 226 | 1 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 {csn 4626 ∪ cuni 4907 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 df-uni 4908 |
| This theorem is referenced by: eusv2nf 5395 |
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