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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 2752 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦) | |
2 | id 22 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴) | |
3 | 1, 2 | nfeqd 2758 | . . 3 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴) |
4 | 3 | alrimiv 1842 | . 2 ⊢ (Ⅎ𝑥𝐴 → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) |
5 | df-nfc 2740 | . . . . 5 ⊢ (Ⅎ𝑥{𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴}) | |
6 | velsn 4141 | . . . . . . 7 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴) | |
7 | 6 | nfbii 1770 | . . . . . 6 ⊢ (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴) |
8 | 7 | albii 1737 | . . . . 5 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) |
9 | 5, 8 | sylbbr 225 | . . . 4 ⊢ (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥{𝐴}) |
10 | 9 | nfunid 4379 | . . 3 ⊢ (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥∪ {𝐴}) |
11 | nfa1 2015 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 𝐴 ∈ 𝑉 | |
12 | unisng 4388 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) | |
13 | 12 | sps 2043 | . . . 4 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) |
14 | 11, 13 | nfceqdf 2747 | . . 3 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥∪ {𝐴} ↔ Ⅎ𝑥𝐴)) |
15 | 10, 14 | syl5ib 233 | . 2 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)) |
16 | 4, 15 | impbid2 215 | 1 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 Ⅎwnfc 2738 {csn 4125 ∪ cuni 4372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-un 3545 df-sn 4126 df-pr 4128 df-uni 4373 |
This theorem is referenced by: eusv2nf 4790 |
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