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Mirrors > Home > MPE Home > Th. List > dffr6 | Structured version Visualization version GIF version |
Description: Alternate definition of df-fr 5630. See dffr5 34712 for a definition without dummy variables (but note that their equivalence uses ax-sep 5298). (Contributed by BJ, 16-Nov-2024.) |
Ref | Expression |
---|---|
dffr6 | ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | velpw 4606 | . . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
2 | 1 | bicomi 223 | . . . . . 6 ⊢ (𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ 𝒫 𝐴) |
3 | velsn 4643 | . . . . . . . 8 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅) | |
4 | 3 | bicomi 223 | . . . . . . 7 ⊢ (𝑥 = ∅ ↔ 𝑥 ∈ {∅}) |
5 | 4 | necon3abii 2987 | . . . . . 6 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 ∈ {∅}) |
6 | 2, 5 | anbi12i 627 | . . . . 5 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅})) |
7 | eldif 3957 | . . . . 5 ⊢ (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅})) | |
8 | 6, 7 | bitr4i 277 | . . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) ↔ 𝑥 ∈ (𝒫 𝐴 ∖ {∅})) |
9 | 8 | imbi1i 349 | . . 3 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) ↔ (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)) |
10 | 9 | albii 1821 | . 2 ⊢ (∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)) |
11 | df-fr 5630 | . 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)) | |
12 | df-ral 3062 | . 2 ⊢ (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦 ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)) | |
13 | 10, 11, 12 | 3bitr4i 302 | 1 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4321 𝒫 cpw 4601 {csn 4627 class class class wbr 5147 Fr wfr 5627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-v 3476 df-dif 3950 df-in 3954 df-ss 3964 df-pw 4603 df-sn 4628 df-fr 5630 |
This theorem is referenced by: frd 5634 |
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