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| Mirrors > Home > MPE Home > Th. List > Mathboxes > eldisjs | Structured version Visualization version GIF version | ||
| Description: Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 24-Jul-2021.) |
| Ref | Expression |
|---|---|
| eldisjs | ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfdisjs 38709 | . 2 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels } | |
| 2 | cnveq 5884 | . . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅) | |
| 3 | 2 | cosseqd 38429 | . . 3 ⊢ (𝑟 = 𝑅 → ≀ ◡𝑟 = ≀ ◡𝑅) |
| 4 | 3 | eleq1d 2826 | . 2 ⊢ (𝑟 = 𝑅 → ( ≀ ◡𝑟 ∈ CnvRefRels ↔ ≀ ◡𝑅 ∈ CnvRefRels )) |
| 5 | 1, 4 | rabeqel 38255 | 1 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels )) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ◡ccnv 5684 ≀ ccoss 38182 Rels crels 38184 CnvRefRels ccnvrefrels 38190 Disjs cdisjs 38215 |
| 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-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-in 3958 df-ss 3968 df-br 5144 df-opab 5206 df-cnv 5693 df-coss 38412 df-disjss 38704 df-disjs 38705 |
| This theorem is referenced by: eldisjs2 38724 eldisjsdisj 38728 |
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